%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SEU909^5 : TPTP v6.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n118.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 13:33:22 EDT 2014 % Result : Theorem 32.93s % Output : Proof 32.93s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SEU909^5 : TPTP v6.1.0. Released v4.0.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n118.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 11:42:21 CDT 2014 % % CPUTime : 32.93 % 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 0x22a0830>, <kernel.Type object at 0x22a0d40>) of role type named a_type % Using role type % Declaring a:Type % FOF formula (<kernel.Constant object at 0x247e5f0>, <kernel.DependentProduct object at 0x22a0518>) of role type named cA % Using role type % Declaring cA:((a->Prop)->Prop) % FOF formula ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) of role conjecture named cDOMTHM8_pme % Conjecture to prove = ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))):Prop % Parameter a_DUMMY:a. % We need to prove ['((and (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))))))'] % Parameter a:Type. % Parameter cA:((a->Prop)->Prop). % Trying to prove ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) % Found x:(cA Xx) % Found (fun (x:(cA Xx))=> x) as proof of (cA Xx) % Found (fun (Xx:(a->Prop)) (x:(cA Xx))=> x) as proof of ((cA Xx)->(cA Xx)) % Found (fun (Xx:(a->Prop)) (x:(cA Xx))=> x) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx))) % Found x1:(cA Xx) % Found (fun (x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x1) as proof of (cA Xx) % Found (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x1) as proof of ((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->(cA Xx)) % Found (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x1) as proof of ((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->(cA Xx))) % Found (and_rect00 (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x1)) as proof of (cA Xx) % Found ((and_rect0 (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x1)) as proof of (cA Xx) % Found (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x1)) as proof of (cA Xx) % Found (fun (x0:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x1))) as proof of (cA Xx) % Found (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x1))) as proof of (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)) % Found (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x1))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx))) % Found x3:(cA Xx) % Found (fun (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3) as proof of (cA Xx) % Found (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3) as proof of ((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->(cA Xx)) % Found (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3) as proof of ((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->(cA Xx))) % Found (and_rect10 (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)) as proof of (cA Xx) % Found ((and_rect1 (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)) as proof of (cA Xx) % Found (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)) as proof of (cA Xx) % Found (fun (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3))) as proof of (cA Xx) % Found (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3))) as proof of (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)) % Found (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx))) % Found eq_ref000:=(eq_ref00 (and (cA Xy))):(((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))) % Found (eq_ref00 (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found ((eq_ref0 (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))) % Found eq_ref00:=(eq_ref0 (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))):(((eq Prop) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) % Found (eq_ref0 (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) b) % Found ((eq_ref Prop) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) b) % Found ((eq_ref Prop) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) b) % Found ((eq_ref Prop) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) b) % Found eq_ref000:=(eq_ref00 (and (cA Xy))):(((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))) % Found (eq_ref00 (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found ((eq_ref0 (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))) % Found eq_ref000:=(eq_ref00 (and (cA Xy))):(((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))) % Found (eq_ref00 (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found ((eq_ref0 (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))) % Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0)) % Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))) % Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) % Found (eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) b) % Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) b) % Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) b) % Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) b) % Found x00:((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) % Instantiate: x1:=Xe:(a->Prop) % Found (fun (x00:((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))))=> x00) as proof of ((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) % Found (fun (Xy:(a->Prop)) (x00:((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))))=> x00) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop)) (x00:((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))))=> x00) as proof of (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))) % Found eq_ref000:=(eq_ref00 (and (cA Xy))):(((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))) % Found (eq_ref00 (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found ((eq_ref0 (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))) % Found x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe))) % Instantiate: x3:=Xe:(a->Prop) % Found x0 as proof of (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3))) % Found x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))) % Instantiate: x3:=Xe:(a->Prop) % Found x2 as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))) % Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0)) % Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0)) % Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0)) % Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0)) % Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0)) % Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0)) % Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))) % Found eq_ref000:=(eq_ref00 (and (cA Xy))):(((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))) % Found (eq_ref00 (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found ((eq_ref0 (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy)))) as proof of (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))) % Found x2:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))) % Found x2 as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))) % Found x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe))) % Found x0 as proof of (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3))) % Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))) % Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop % Found eq_sym0 as proof of b % Found x3:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))) % Instantiate: x1:=Xe:(a->Prop) % Found x3 as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))) % Found x1:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe))) % Instantiate: x3:=Xe:(a->Prop) % Found x1 as proof of (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3))) % Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) % Found (eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) b) % Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) b) % Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) b) % Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) b) % Found x3:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))) % Found x3 as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))) % Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0)) % Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0)) % Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0)) % Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0)) % Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0)) % Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0)) % Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))) % Found x1:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe))) % Found x1 as proof of (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3))) % Found x3:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))) % Instantiate: x1:=Xe:(a->Prop) % Found x3 as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))) % Found x3:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))) % Found x3 as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))) % Found x2:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe))) % Instantiate: x1:=Xe:(a->Prop) % Found x2 as proof of (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1))) % Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0)) % Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))) % Found x2:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe))) % Found x2 as proof of (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1))) % Found x00:((and (cA Xy)) (forall (Xx0:a), ((x5 Xx0)->(Xy Xx0)))) % Instantiate: x5:=Xe:(a->Prop) % Found (fun (x00:((and (cA Xy)) (forall (Xx0:a), ((x5 Xx0)->(Xy Xx0)))))=> x00) as proof of ((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) % Found (fun (Xy:(a->Prop)) (x00:((and (cA Xy)) (forall (Xx0:a), ((x5 Xx0)->(Xy Xx0)))))=> x00) as proof of (((and (cA Xy)) (forall (Xx0:a), ((x5 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))) % Found (fun (Xy:(a->Prop)) (x00:((and (cA Xy)) (forall (Xx0:a), ((x5 Xx0)->(Xy Xx0)))))=> x00) as proof of (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x5 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))) % Found eq_ref00:=(eq_ref0 (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))):(((eq ((a->Prop)->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) % Found (eq_ref0 (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) b) % Found ((eq_ref ((a->Prop)->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) b) % Found ((eq_ref ((a->Prop)->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) b) % Found ((eq_ref ((a->Prop)->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) b) % Found x2:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe))) % Instantiate: x1:=Xe:(a->Prop) % Found x2 as proof of (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1))) % Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0)) % Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0)) % Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0))) % Found x2:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe))) % Found x2 as proof of (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x1 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x1))) % Found conj300:=(conj30 (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))):((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) % Found (conj30 (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) as proof of ((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) % Found ((fun (B:Prop)=> ((conj3 B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) as proof of ((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) % Found ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) as proof of ((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) % Found (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) as proof of ((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) % Found (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) as proof of ((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))))) % Found (and_rect10 (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))) as proof of ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))) % Found ((and_rect1 ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))) as proof of ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))) % Found (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))) as proof of ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))) % Found (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))) as proof of ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))) % Found ((conj20 (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))))) as proof of ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) % Found (((conj2 (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))))) as proof of ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) % Found ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))))) as proof of ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) % Found ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))))) as proof of ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) % Found (ex_intro000 ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x3 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x3)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy)))))) as proof of ((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) % Found ((ex_intro00 Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))) as proof of ((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) % Found (((ex_intro0 (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))) as proof of ((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) % Found ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))) as proof of ((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) % Found (fun (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy))))))) as proof of ((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) % Found (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy))))))) as proof of (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))) % Found (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy))))))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))) % Found ((conj10 (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) % Found (((conj1 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) % Found ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) % Found (fun (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy))))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) % Found (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy))))))))) as proof of ((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) % Found (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy))))))))) as proof of ((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))))) % Found (and_rect00 (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) % Found ((and_rect0 ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) % Found (((fun (P:Type) (x0:((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->P)))=> (((((and_rect (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))) P) x0) x)) ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) % Found (fun (x:((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))))=> (((fun (P:Type) (x0:((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->P)))=> (((((and_rect (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))) P) x0) x)) ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy))))))))))) as proof of ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) % Found (fun (Xe:(a->Prop)) (x:((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))))=> (((fun (P:Type) (x0:((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->P)))=> (((((and_rect (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))) P) x0) x)) ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy))))))))))) as proof of (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) % Found (fun (Xe:(a->Prop)) (x:((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))))=> (((fun (P:Type) (x0:((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->P)))=> (((((and_rect (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))) P) x0) x)) ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy))))))))))) as proof of (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))))) % Found ((conj00 (fun (Xx:(a->Prop)) (x:(cA Xx))=> x)) (fun (Xe:(a->Prop)) (x:((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))))=> (((fun (P:Type) (x0:((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->P)))=> (((((and_rect (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))) P) x0) x)) ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))))) as proof of ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) % Found (((conj0 (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) (fun (Xx:(a->Prop)) (x:(cA Xx))=> x)) (fun (Xe:(a->Prop)) (x:((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))))=> (((fun (P:Type) (x0:((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->P)))=> (((((and_rect (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))) P) x0) x)) ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))))) as proof of ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) % Found ((((conj (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) (fun (Xx:(a->Prop)) (x:(cA Xx))=> x)) (fun (Xe:(a->Prop)) (x:((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))))=> (((fun (P:Type) (x0:((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->P)))=> (((((and_rect (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))) P) x0) x)) ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))))) as proof of ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) % Found ((((conj (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) (fun (Xx:(a->Prop)) (x:(cA Xx))=> x)) (fun (Xe:(a->Prop)) (x:((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))))=> (((fun (P:Type) (x0:((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->P)))=> (((((and_rect (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))) P) x0) x)) ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))))) as proof of ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) % Got proof ((((conj (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) (fun (Xx:(a->Prop)) (x:(cA Xx))=> x)) (fun (Xe:(a->Prop)) (x:((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))))=> (((fun (P:Type) (x0:((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->P)))=> (((((and_rect (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))) P) x0) x)) ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))))) % Time elapsed = 32.414753s % node=2503 cost=1277.000000 depth=35 % :::::::::::::::::::::: % % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % ((((conj (forall (Xx:(a->Prop)), ((cA Xx)->(cA Xx)))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))))) (fun (Xx:(a->Prop)) (x:(cA Xx))=> x)) (fun (Xe:(a->Prop)) (x:((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))))=> (((fun (P:Type) (x0:((forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))->((forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))->P)))=> (((((and_rect (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx))))))) P) x0) x)) ((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))))))) (fun (x0:(forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (x1:(forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))=> ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))->((ex (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))))))))))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))=> x3)))) (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))))=> ((((ex_intro (a->Prop)) (fun (Xe_0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe_0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe_0)))) (forall (Xx0:a), ((Xe_0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe_0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))))) Xe) ((((conj ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0))))))) (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0)))) P) x4) x2)) ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))) (fun (x4:(cA Xx))=> ((fun (B:Prop)=> (((conj (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) B) x0)) (forall (Xx0:a), ((Xe Xx0)->(Xx Xx0))))))) (fun (Xy:(a->Prop))=> (((eq_ref Prop) (forall (Xx0:a), ((Xe Xx0)->(Xy Xx0)))) (and (cA Xy)))))))))))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------