TSTP Solution File: SEU890^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU890^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n092.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:20 EDT 2014
% Result : Theorem 105.04s
% Output : Proof 105.04s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU890^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n092.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:40:56 CDT 2014
% % CPUTime : 105.04
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0xd71cf8>, <kernel.Type object at 0xd71a70>) of role type named c_type
% Using role type
% Declaring c:Type
% FOF formula (<kernel.Constant object at 0x11a2bd8>, <kernel.Type object at 0xd71d40>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0xd71b48>, <kernel.Type object at 0xd71b90>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0xd71a70>, <kernel.DependentProduct object at 0xd71560>) of role type named cG
% Using role type
% Declaring cG:(c->b)
% FOF formula (<kernel.Constant object at 0xd71d40>, <kernel.DependentProduct object at 0xd71518>) of role type named cF
% Using role type
% Declaring cF:(b->a)
% FOF formula (<kernel.Constant object at 0xd71680>, <kernel.DependentProduct object at 0xd71ef0>) of role type named cS
% Using role type
% Declaring cS:(c->Prop)
% FOF formula (forall (Xx:a), ((iff ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))) of role conjecture named cTHM29_pme
% Conjecture to prove = (forall (Xx:a), ((iff ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))):Prop
% Parameter c_DUMMY:c.
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xx:a), ((iff ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))))']
% Parameter c:Type.
% Parameter b:Type.
% Parameter a:Type.
% Parameter cG:(c->b).
% Parameter cF:(b->a).
% Parameter cS:(c->Prop).
% Trying to prove (forall (Xx:a), ((iff ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))):(((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) (fun (x:c)=> ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found (eta_expansion00 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found (fun (x0:c)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found (fun (x0:c)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found (fun (x0:c)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found (fun (x0:c)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))):(((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) (fun (x:c)=> ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found (eta_expansion00 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found x3:(((eq a) Xx) (cF (cG x0)))
% Instantiate: x4:=(cG x0):b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x4:(((eq a) Xx) (cF (cG x1)))
% Instantiate: x0:=(cG x1):b
% Found x4 as proof of (((eq a) Xx) (cF x0))
% Found x4:(((eq a) Xx) (cF (cG x0)))
% Instantiate: x2:=(cG x0):b
% Found x4 as proof of (((eq a) Xx) (cF x2))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found (fun (x0:c)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found (fun (x0:c)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found (fun (x0:c)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))
% Found (fun (x0:c)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))):(((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) (fun (x:c)=> ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found (eta_expansion00 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found (eta_expansion00 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found x3:(((eq a) Xx) (cF (cG x0)))
% Instantiate: x4:=(cG x0):b
% Found x3 as proof of (((eq a) Xx) (cF x4))
% Found x4:(((eq a) Xx) (cF (cG x1)))
% Instantiate: x0:=(cG x1):b
% Found x4 as proof of (((eq a) Xx) (cF x0))
% Found x4:(((eq a) Xx) (cF (cG x0)))
% Instantiate: x2:=(cG x0):b
% Found x4 as proof of (((eq a) Xx) (cF x2))
% Found x4:(((eq a) Xx) (cF (cG x1)))
% Instantiate: x0:=(cG x1):b
% Found (fun (x4:(((eq a) Xx) (cF (cG x1))))=> x4) as proof of (((eq a) Xx) (cF x0))
% Found (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4) as proof of ((cS x1)->((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0))))
% Found (and_rect00 (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF (cG x0))))
% Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF x2)))
% Found ((eq_substitution000 (cF (cG x0))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF x2)))
% Found (((eq_substitution00 Xx) (cF (cG x0))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF x2)))
% Found ((((eq_substitution0 a) Xx) (cF (cG x0))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF x2)))
% Found (((((eq_substitution a) a) Xx) (cF (cG x0))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF x2)))
% Found (((((eq_substitution a) a) Xx) (cF (cG x0))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF x2)))
% Found (fun (x3:(cS x0))=> (((((eq_substitution a) a) Xx) (cF (cG x0))) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF x2)))
% Found (fun (x3:(cS x0))=> (((((eq_substitution a) a) Xx) (cF (cG x0))) (fun (x6:a)=> x6))) as proof of ((cS x0)->((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF x2))))
% Found (and_rect00 (fun (x3:(cS x0))=> (((((eq_substitution a) a) Xx) (cF (cG x0))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:(cS x0))=> (((((eq_substitution a) a) Xx) (cF (cG x0))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:(cS x0))=> (((((eq_substitution a) a) Xx) (cF (cG x0))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:(cS x0))=> (((((eq_substitution a) a) Xx) (cF (cG x0))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x2))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found (fun (x2:c)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found (fun (x2:c)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found (fun (x2:c)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found (fun (x2:c)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))):(((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (eq_ref0 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eq_ref (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eq_ref (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eq_ref (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: b0:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found x2 as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))):(((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) (fun (x:c)=> ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found (eta_expansion00 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: b0:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P b0)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P b0))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P b0)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found ((and_rect0 (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: b0:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P b0)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P b0))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P b0)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found ((and_rect0 (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found (fun (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2))) as proof of (P b0)
% Found x4:(((eq a) Xx) (cF (cG x1)))
% Instantiate: x0:=(cG x1):b
% Found (fun (x4:(((eq a) Xx) (cF (cG x1))))=> x4) as proof of (((eq a) Xx) (cF x0))
% Found (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4) as proof of ((cS x1)->((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0))))
% Found (and_rect00 (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4)) as proof of (((eq a) Xx) (cF x0))
% Found (fun (x2:((and (cS x1)) (((eq a) Xx) (cF (cG x1)))))=> (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:(cS x1)) (x4:(((eq a) Xx) (cF (cG x1))))=> x4))) as proof of (((eq a) Xx) (cF x0))
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found x2 as proof of (P f)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found x2 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x2) (cG Xt0)))))) (((eq a) Xx) (cF x2))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found (fun (x2:c)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found (fun (x2:c)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found (fun (x2:c)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and (cS x2)) (((eq a) Xx) (cF (cG x2)))))
% Found (fun (x2:c)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF (cG x1))))
% Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found ((eq_substitution000 (cF (cG x1))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (((eq_substitution00 Xx) (cF (cG x1))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found ((((eq_substitution0 a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6))) as proof of ((cS x1)->((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0))))
% Found (and_rect00 (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found x4:(((eq a) Xx) (cF (cG x0)))
% Instantiate: x2:=(cG x0):b
% Found (fun (x4:(((eq a) Xx) (cF (cG x0))))=> x4) as proof of (((eq a) Xx) (cF x2))
% Found (fun (x3:(cS x0)) (x4:(((eq a) Xx) (cF (cG x0))))=> x4) as proof of ((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF x2)))
% Found (fun (x3:(cS x0)) (x4:(((eq a) Xx) (cF (cG x0))))=> x4) as proof of ((cS x0)->((((eq a) Xx) (cF (cG x0)))->(((eq a) Xx) (cF x2))))
% Found (and_rect00 (fun (x3:(cS x0)) (x4:(((eq a) Xx) (cF (cG x0))))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found ((and_rect0 (((eq a) Xx) (cF x2))) (fun (x3:(cS x0)) (x4:(((eq a) Xx) (cF (cG x0))))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:(cS x0)) (x4:(((eq a) Xx) (cF (cG x0))))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found (((fun (P:Type) (x3:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x3) x1)) (((eq a) Xx) (cF x2))) (fun (x3:(cS x0)) (x4:(((eq a) Xx) (cF (cG x0))))=> x4)) as proof of (((eq a) Xx) (cF x2))
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P f)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P f))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P f)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P f)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P f))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P f)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P f)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P f))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P f)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (fun (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2))) as proof of (P f)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P f)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P f))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P f)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (fun (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2))) as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (fun (x:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found (eta_expansion_dep00 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found (fun (x4:c)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found (fun (x4:c)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found (fun (x4:c)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found (fun (x4:c)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: b0:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found x2 as proof of (P b0)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: b0:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P b0)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P b0))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P b0)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found ((and_rect0 (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))):(((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) (fun (x:c)=> ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found (eta_expansion00 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found (((eta_expansion c) Prop) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: b0:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P b0)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P b0))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P b0)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found ((and_rect0 (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P b0)
% Found (fun (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P b0)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2))) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eq_substitution00000:=(eq_substitution0000 (fun (x6:a)=> x6)):((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF (cG x1))))
% Found (eq_substitution0000 (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found ((eq_substitution000 (cF (cG x1))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (((eq_substitution00 Xx) (cF (cG x1))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found ((((eq_substitution0 a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6))) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0)))
% Found (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6))) as proof of ((cS x1)->((((eq a) Xx) (cF (cG x1)))->(((eq a) Xx) (cF x0))))
% Found (and_rect00 (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found ((and_rect0 (((eq a) Xx) (cF x0))) (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6)))) as proof of (((eq a) Xx) (cF x0))
% Found (fun (x2:((and (cS x1)) (((eq a) Xx) (cF (cG x1)))))=> (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) Xx) (cF x0))) (fun (x3:(cS x1))=> (((((eq_substitution a) a) Xx) (cF (cG x1))) (fun (x6:a)=> x6))))) as proof of (((eq a) Xx) (cF x0))
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found x2 as proof of (P f)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found x2 as proof of (P f)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P f)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P f))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P f)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P f)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P f))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P f)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P f)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P f))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P f)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (fun (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2))) as proof of (P f)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (P f)
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of ((((eq a) Xx) (cF x0))->(P f))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->(P f)))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2)) as proof of (P f)
% Found (fun (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P0:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P0)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P0) x2) x1)) (P f)) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> x2))) as proof of (P f)
% Found eq_ref00:=(eq_ref0 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))):(((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (eq_ref0 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found ((eq_ref (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) as proof of (((eq (b->Prop)) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) b0)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found (fun (x4:c)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found (fun (x4:c)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found (fun (x4:c)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and (cS x4)) (((eq a) Xx) (cF (cG x4)))))
% Found (fun (x4:c)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref000:=(eq_ref00 P):((P Xx)->(P Xx))
% Found (eq_ref00 P) as proof of (P0 Xx)
% Found ((eq_ref0 Xx) P) as proof of (P0 Xx)
% Found (((eq_ref a) Xx) P) as proof of (P0 Xx)
% Found (((eq_ref a) Xx) P) as proof of (P0 Xx)
% Found eq_ref000:=(eq_ref00 P):((P Xx)->(P Xx))
% Found (eq_ref00 P) as proof of (P0 Xx)
% Found ((eq_ref0 Xx) P) as proof of (P0 Xx)
% Found (((eq_ref a) Xx) P) as proof of (P0 Xx)
% Found (((eq_ref a) Xx) P) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) (cF (cG x0)))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF (cG x0)))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF (cG x0)))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF (cG x0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) (cF x0))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF x0))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF x0))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF x0))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: b0:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found x2 as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))):(((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (eq_ref0 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eq_ref (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eq_ref (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found ((eq_ref (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) as proof of (((eq (c->Prop)) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) b0)
% Found x6:(cS x4)
% Instantiate: x8:=x4:c
% Found x6 as proof of (cS x8)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))))
% Found (fun (x4:b)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:b), (((eq Prop) (f x)) ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))))
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found x2 as proof of (P f)
% Found x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))
% Instantiate: f:=(fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))):(c->Prop)
% Found x2 as proof of (P f)
% Found x7:(cS x5)
% Instantiate: x0:=x5:c
% Found x7 as proof of (cS x0)
% Found x7:(cS x5)
% Instantiate: x2:=x5:c
% Found x7 as proof of (cS x2)
% Found x7:(cS x5)
% Instantiate: x4:=x5:c
% Found x7 as proof of (cS x4)
% Found x7:(cS x4)
% Instantiate: x6:=x4:c
% Found x7 as proof of (cS x6)
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and (cS x6)) (((eq a) Xx) (cF (cG x6)))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and (cS x6)) (((eq a) Xx) (cF (cG x6)))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and (cS x6)) (((eq a) Xx) (cF (cG x6)))))
% Found (fun (x6:c)=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and (cS x6)) (((eq a) Xx) (cF (cG x6)))))
% Found (fun (x6:c)=> ((eq_ref Prop) (f x6))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((and (cS x6)) (((eq a) Xx) (cF (cG x6)))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and (cS x6)) (((eq a) Xx) (cF (cG x6)))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((and (cS x6)) (((eq a) Xx) (cF (cG x6)))))
% Found (fun (x6:c)=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((and (cS x6)) (((eq a) Xx) (cF (cG x6)))))
% Found (fun (x6:c)=> ((eq_ref Prop) (f x6))) as proof of (forall (x:c), (((eq Prop) (f x)) ((and (cS x)) (((eq a) Xx) (cF (cG x))))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) (cF (cG x0)))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF (cG x0)))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF (cG x0)))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF (cG x0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) (cF x0))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF x0))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF x0))
% Found ((eq_ref a) b0) as proof of (((eq a) b0) (cF x0))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref000:=(eq_ref00 P):((P Xx)->(P Xx))
% Found (eq_ref00 P) as proof of (P0 Xx)
% Found ((eq_ref0 Xx) P) as proof of (P0 Xx)
% Found (((eq_ref a) Xx) P) as proof of (P0 Xx)
% Found (((eq_ref a) Xx) P) as proof of (P0 Xx)
% Found eq_ref000:=(eq_ref00 P):((P Xx)->(P Xx))
% Found (eq_ref00 P) as proof of (P0 Xx)
% Found ((eq_ref0 Xx) P) as proof of (P0 Xx)
% Found (((eq_ref a) Xx) P) as proof of (P0 Xx)
% Found (((eq_ref a) Xx) P) as proof of (P0 Xx)
% Found x700:=(x70 x3):(((eq a) Xx) (cF (cG x4)))
% Found (x70 x3) as proof of (((eq a) Xx) (cF (cG x8)))
% Found ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3) as proof of (((eq a) Xx) (cF (cG x8)))
% Found ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3) as proof of (((eq a) Xx) (cF (cG x8)))
% Found ((conj10 x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)) as proof of ((and (cS x8)) (((eq a) Xx) (cF (cG x8))))
% Found (((conj1 (((eq a) Xx) (cF (cG x8)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)) as proof of ((and (cS x8)) (((eq a) Xx) (cF (cG x8))))
% Found ((((conj (cS x8)) (((eq a) Xx) (cF (cG x8)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)) as proof of ((and (cS x8)) (((eq a) Xx) (cF (cG x8))))
% Found ((((conj (cS x8)) (((eq a) Xx) (cF (cG x8)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)) as proof of ((and (cS x8)) (((eq a) Xx) (cF (cG x8))))
% Found (ex_intro000 ((((conj (cS x8)) (((eq a) Xx) (cF (cG x8)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found ((ex_intro00 x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (((ex_intro0 (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (fun (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))) as proof of ((((eq b) x0) (cG x4))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))
% Found (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))) as proof of ((cS x4)->((((eq b) x0) (cG x4))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))))
% Found (and_rect10 (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found ((and_rect1 ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (fun (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))) as proof of (((and (cS x4)) (((eq b) x0) (cG x4)))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))
% Found (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))) as proof of (forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))))
% Found (ex_ind10 (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found ((ex_ind1 ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (fun (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))))) as proof of ((((eq a) Xx) (cF x0))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))
% Found (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))))) as proof of (((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))))
% Found (and_rect00 (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found ((and_rect0 ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (fun (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))))))) as proof of (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0)))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))
% Found (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))))))) as proof of (forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))))
% Found (ex_ind00 (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found ((ex_ind0 ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x:c), (((and (cS x)) (((eq b) x0) (cG x)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (((fun (P:Prop) (x0:(forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x1:c), (((and (cS x1)) (((eq b) x0) (cG x1)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (fun (x:((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x1:c), (((and (cS x1)) (((eq b) x0) (cG x1)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))))))))) as proof of ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))
% Found (fun (x:((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x1:c), (((and (cS x1)) (((eq b) x0) (cG x1)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3)))))))))))) as proof of (((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))
% Found eq_sym010:=(eq_sym01 (cF (cG x1))):((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF (cG x1))) Xx))
% Found (eq_sym01 (cF (cG x1))) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF x0)) Xx))
% Found ((eq_sym0 Xx) (cF (cG x1))) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF x0)) Xx))
% Found ((eq_sym0 Xx) (cF (cG x1))) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF x0)) Xx))
% Found (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1)))) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF x0)) Xx))
% Found (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1)))) as proof of ((cS x1)->((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF x0)) Xx)))
% Found (and_rect00 (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1))))) as proof of (((eq a) (cF x0)) Xx)
% Found ((and_rect0 (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1))))) as proof of (((eq a) (cF x0)) Xx)
% Found (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1))))) as proof of (((eq a) (cF x0)) Xx)
% Found (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1))))) as proof of (((eq a) (cF x0)) Xx)
% Found (eq_sym000 (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1)))))) as proof of (((eq a) Xx) (cF x0))
% Found ((eq_sym00 Xx) (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1)))))) as proof of (((eq a) Xx) (cF x0))
% Found (((eq_sym0 (cF x0)) Xx) (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1)))))) as proof of (((eq a) Xx) (cF x0))
% Found ((((eq_sym a) (cF x0)) Xx) (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> (((eq_sym a) Xx) (cF (cG x1)))))) as proof of (((eq a) Xx) (cF x0))
% Found (fun (x2:((and (cS x1)) (((eq a) Xx) (cF (cG x1)))))=> ((((eq_sym a) (cF x0)) Xx) (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> (((eq_sym a) Xx) (cF (cG x1))))))) as proof of (((eq a) Xx) (cF x0))
% Found eq_ref00:=(eq_ref0 x2):(((eq b) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq b) x2) (cG x3))
% Found ((eq_ref b) x2) as proof of (((eq b) x2) (cG x3))
% Found ((eq_ref b) x2) as proof of (((eq b) x2) (cG x3))
% Found ((eq_ref b) x2) as proof of (((eq b) x2) (cG x3))
% Found eq_sym010:=(eq_sym01 (cF (cG x1))):((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF (cG x1))) Xx))
% Found (eq_sym01 (cF (cG x1))) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF x0)) Xx))
% Found ((eq_sym0 Xx) (cF (cG x1))) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF x0)) Xx))
% Found ((eq_sym0 Xx) (cF (cG x1))) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF x0)) Xx))
% Found (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1)))) as proof of ((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF x0)) Xx))
% Found (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1)))) as proof of ((cS x1)->((((eq a) Xx) (cF (cG x1)))->(((eq a) (cF x0)) Xx)))
% Found (and_rect00 (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1))))) as proof of (((eq a) (cF x0)) Xx)
% Found ((and_rect0 (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1))))) as proof of (((eq a) (cF x0)) Xx)
% Found (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1))))) as proof of (((eq a) (cF x0)) Xx)
% Found (fun (x2:((and (cS x1)) (((eq a) Xx) (cF (cG x1)))))=> (((fun (P:Type) (x3:((cS x1)->((((eq a) Xx) (cF (cG x1)))->P)))=> (((((and_rect (cS x1)) (((eq a) Xx) (cF (cG x1)))) P) x3) x2)) (((eq a) (cF x0)) Xx)) (fun (x3:(cS x1))=> ((eq_sym0 Xx) (cF (cG x1)))))) as proof of (((eq a) (cF x0)) Xx)
% Found eq_ref00:=(eq_ref0 x2):(((eq b) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq b) x2) (cG x3))
% Found ((eq_ref b) x2) as proof of (((eq b) x2) (cG x3))
% Found ((eq_ref b) x2) as proof of (((eq b) x2) (cG x3))
% Found ((eq_ref b) x2) as proof of (((eq b) x2) (cG x3))
% Found eq_ref00:=(eq_ref0 x0):(((eq b) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq b) x0) (cG x3))
% Found ((eq_ref b) x0) as proof of (((eq b) x0) (cG x3))
% Found ((eq_ref b) x0) as proof of (((eq b) x0) (cG x3))
% Found ((eq_ref b) x0) as proof of (((eq b) x0) (cG x3))
% Found eq_ref00:=(eq_ref0 x0):(((eq b) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq b) x0) (cG x3))
% Found ((eq_ref b) x0) as proof of (((eq b) x0) (cG x3))
% Found ((eq_ref b) x0) as proof of (((eq b) x0) (cG x3))
% Found ((eq_ref b) x0) as proof of (((eq b) x0) (cG x3))
% Found eq_ref00:=(eq_ref0 x0):(((eq b) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq b) x0) (cG x1))
% Found ((eq_ref b) x0) as proof of (((eq b) x0) (cG x1))
% Found ((eq_ref b) x0) as proof of (((eq b) x0) (cG x1))
% Found ((eq_ref b) x0) as proof of (((eq b) x0) (cG x1))
% Found x2:(cS x0)
% Instantiate: x5:=x0:c
% Found x2 as proof of (cS x5)
% Found eq_ref00:=(eq_ref0 x4):(((eq b) x4) x4)
% Found (eq_ref0 x4) as proof of (((eq b) x4) (cG x5))
% Found ((eq_ref b) x4) as proof of (((eq b) x4) (cG x5))
% Found ((eq_ref b) x4) as proof of (((eq b) x4) (cG x5))
% Found ((eq_ref b) x4) as proof of (((eq b) x4) (cG x5))
% Found ((conj20 x2) ((eq_ref b) x4)) as proof of ((and (cS x5)) (((eq b) x4) (cG x5)))
% Found (((conj2 (((eq b) x4) (cG x5))) x2) ((eq_ref b) x4)) as proof of ((and (cS x5)) (((eq b) x4) (cG x5)))
% Found ((((conj (cS x5)) (((eq b) x4) (cG x5))) x2) ((eq_ref b) x4)) as proof of ((and (cS x5)) (((eq b) x4) (cG x5)))
% Found ((((conj (cS x5)) (((eq b) x4) (cG x5))) x2) ((eq_ref b) x4)) as proof of ((and (cS x5)) (((eq b) x4) (cG x5)))
% Found (ex_intro100 ((((conj (cS x5)) (((eq b) x4) (cG x5))) x2) ((eq_ref b) x4))) as proof of ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))
% Found ((ex_intro10 x0) ((((conj (cS x0)) (((eq b) x4) (cG x0))) x2) ((eq_ref b) x4))) as proof of ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))
% Found (((ex_intro1 (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) x4) (cG x0))) x2) ((eq_ref b) x4))) as proof of ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))
% Found ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) x4) (cG x0))) x2) ((eq_ref b) x4))) as proof of ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))
% Found ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) x4) (cG x0))) x2) ((eq_ref b) x4))) as proof of ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))
% Found ((conj10 ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) x4) (cG x0))) x2) ((eq_ref b) x4)))) x3) as proof of ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4)))
% Found (((conj1 (((eq a) Xx) (cF x4))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) x4) (cG x0))) x2) ((eq_ref b) x4)))) x3) as proof of ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4)))
% Found ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) x4) (cG x0))) x2) ((eq_ref b) x4)))) x3) as proof of ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4)))
% Found ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) x4) (cG x0))) x2) ((eq_ref b) x4)))) x3) as proof of ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4)))
% Found (ex_intro000 ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0)))))) (((eq a) Xx) (cF x4))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x4) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) x4) (cG x0))) x2) ((eq_ref b) x4)))) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found ((ex_intro00 (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (((ex_intro0 (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (fun (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3))) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3))) as proof of ((((eq a) Xx) (cF (cG x0)))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))
% Found (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3))) as proof of ((cS x0)->((((eq a) Xx) (cF (cG x0)))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))))
% Found (and_rect00 (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found ((and_rect0 ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (fun (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3))))) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3))))) as proof of (((and (cS x0)) (((eq a) Xx) (cF (cG x0))))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))
% Found (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3))))) as proof of (forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))))
% Found (ex_ind00 (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))))) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found ((ex_ind0 ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))))) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (((fun (P:Prop) (x0:(forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->P)))=> (((((ex_ind c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) P) x0) x)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))))) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (fun (x:((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))=> (((fun (P:Prop) (x0:(forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->P)))=> (((((ex_ind c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) P) x0) x)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3))))))) as proof of ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))
% Found (fun (x:((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))=> (((fun (P:Prop) (x0:(forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->P)))=> (((((ex_ind c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) P) x0) x)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3))))))) as proof of (((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))
% Found ((conj00 (fun (x:((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x1:c), (((and (cS x1)) (((eq b) x0) (cG x1)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))))))) (fun (x:((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))=> (((fun (P:Prop) (x0:(forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->P)))=> (((((ex_ind c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) P) x0) x)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))))))) as proof of ((iff ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))
% Found (((conj0 (((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))) (fun (x:((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x1:c), (((and (cS x1)) (((eq b) x0) (cG x1)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))))))) (fun (x:((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))=> (((fun (P:Prop) (x0:(forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->P)))=> (((((ex_ind c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) P) x0) x)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))))))) as proof of ((iff ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))
% Found ((((conj (((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))) (((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))) (fun (x:((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x1:c), (((and (cS x1)) (((eq b) x0) (cG x1)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))))))) (fun (x:((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))=> (((fun (P:Prop) (x0:(forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->P)))=> (((((ex_ind c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) P) x0) x)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))))))) as proof of ((iff ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))
% Found (fun (Xx:a)=> ((((conj (((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))) (((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))) (fun (x:((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x1:c), (((and (cS x1)) (((eq b) x0) (cG x1)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))))))) (fun (x:((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))=> (((fun (P:Prop) (x0:(forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->P)))=> (((((ex_ind c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) P) x0) x)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3))))))))) as proof of ((iff ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))
% Found (fun (Xx:a)=> ((((conj (((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))) (((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))) (fun (x:((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x1:c), (((and (cS x1)) (((eq b) x0) (cG x1)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))))))) (fun (x:((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))=> (((fun (P:Prop) (x0:(forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->P)))=> (((((ex_ind c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) P) x0) x)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3))))))))) as proof of (forall (Xx:a), ((iff ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))))
% Got proof (fun (Xx:a)=> ((((conj (((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))) (((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))) (fun (x:((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x1:c), (((and (cS x1)) (((eq b) x0) (cG x1)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))))))) (fun (x:((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))=> (((fun (P:Prop) (x0:(forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->P)))=> (((((ex_ind c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) P) x0) x)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))))))))
% Time elapsed = 103.873640s
% node=12680 cost=1830.000000 depth=39
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (Xx:a)=> ((((conj (((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))->((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))) (((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))->((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))) (fun (x:((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))))=> (((fun (P:Prop) (x0:(forall (x:b), (((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x) (cG Xt0)))))) (((eq a) Xx) (cF x)))->P)))=> (((((ex_ind b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) P) x0) x)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x0:b) (x1:((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))))=> (((fun (P:Type) (x2:(((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))->((((eq a) Xx) (cF x0))->P)))=> (((((and_rect ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (((eq a) Xx) (cF x0))) P) x2) x1)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x2:((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0)))))) (x3:(((eq a) Xx) (cF x0)))=> (((fun (P:Prop) (x4:(forall (x1:c), (((and (cS x1)) (((eq b) x0) (cG x1)))->P)))=> (((((ex_ind c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) x0) (cG Xt0))))) P) x4) x2)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x4:c) (x5:((and (cS x4)) (((eq b) x0) (cG x4))))=> (((fun (P:Type) (x6:((cS x4)->((((eq b) x0) (cG x4))->P)))=> (((((and_rect (cS x4)) (((eq b) x0) (cG x4))) P) x6) x5)) ((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt))))))) (fun (x6:(cS x4)) (x7:(((eq b) x0) (cG x4)))=> ((((ex_intro c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) x4) ((((conj (cS x4)) (((eq a) Xx) (cF (cG x4)))) x6) ((x7 (fun (x10:b)=> (((eq a) Xx) (cF x10)))) x3))))))))))))) (fun (x:((ex c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))))=> (((fun (P:Prop) (x0:(forall (x:c), (((and (cS x)) (((eq a) Xx) (cF (cG x))))->P)))=> (((((ex_ind c) (fun (Xt:c)=> ((and (cS Xt)) (((eq a) Xx) (cF (cG Xt)))))) P) x0) x)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x0:c) (x1:((and (cS x0)) (((eq a) Xx) (cF (cG x0)))))=> (((fun (P:Type) (x2:((cS x0)->((((eq a) Xx) (cF (cG x0)))->P)))=> (((((and_rect (cS x0)) (((eq a) Xx) (cF (cG x0)))) P) x2) x1)) ((ex b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt)))))) (fun (x2:(cS x0)) (x3:(((eq a) Xx) (cF (cG x0))))=> ((((ex_intro b) (fun (Xt:b)=> ((and ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) Xt) (cG Xt0)))))) (((eq a) Xx) (cF Xt))))) (cG x0)) ((((conj ((ex c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0)))))) (((eq a) Xx) (cF (cG x0)))) ((((ex_intro c) (fun (Xt0:c)=> ((and (cS Xt0)) (((eq b) (cG x0)) (cG Xt0))))) x0) ((((conj (cS x0)) (((eq b) (cG x0)) (cG x0))) x2) ((eq_ref b) (cG x0))))) x3)))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------