%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU852^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n112.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 13:33:17 EDT 2014

% Result   : Unknown 29.22s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU852^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n112.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 11:36:41 CDT 2014
% % CPUTime  : 29.22 
% 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 0x162b9e0>, <kernel.Type object at 0x1b17e18>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (S:(a->Prop)) (T:(a->Prop)) (U:(a->Prop)), (((and (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))->((iff (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))))) of role conjecture named cGAZING_THM36_pme
% Conjecture to prove = (forall (S:(a->Prop)) (T:(a->Prop)) (U:(a->Prop)), (((and (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))->((iff (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (S:(a->Prop)) (T:(a->Prop)) (U:(a->Prop)), (((and (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))->((iff (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))))']
% Parameter a:Type.
% Trying to prove (forall (S:(a->Prop)) (T:(a->Prop)) (U:(a->Prop)), (((and (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))->((iff (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))))
% Found x000:=(x00 x4):(T Xx)
% Found (x00 x4) as proof of (T Xx)
% Found ((x0 Xx) x4) as proof of (T Xx)
% Found (fun (x5:(U Xx))=> ((x0 Xx) x4)) as proof of (T Xx)
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((U Xx)->(T Xx))
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (fun (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (T Xx)
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect00 (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (fun (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x010:=(x01 x3):(T Xx)
% Found (x01 x3) as proof of (T Xx)
% Found ((x0 Xx) x3) as proof of (T Xx)
% Found (fun (x4:(U Xx))=> ((x0 Xx) x3)) as proof of (T Xx)
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((U Xx)->(T Xx))
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (fun (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (T Xx)
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect00 (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x000:=(x00 x4):(T Xx)
% Found (x00 x4) as proof of (T Xx)
% Found ((x0 Xx) x4) as proof of (T Xx)
% Found (fun (x5:(U Xx))=> ((x0 Xx) x4)) as proof of (T Xx)
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((U Xx)->(T Xx))
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (fun (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (T Xx)
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect00 (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (fun (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x3):(T Xx)
% Found (x01 x3) as proof of (T Xx)
% Found ((x0 Xx) x3) as proof of (T Xx)
% Found (fun (x4:(U Xx))=> ((x0 Xx) x3)) as proof of (T Xx)
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((U Xx)->(T Xx))
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (fun (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (T Xx)
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect00 (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found x010:=(x01 x3):(T Xx)
% Found (x01 x3) as proof of (T Xx)
% Found ((x0 Xx) x3) as proof of (T Xx)
% Found (fun (x4:(U Xx))=> ((x0 Xx) x3)) as proof of (T Xx)
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((U Xx)->(T Xx))
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (fun (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (T Xx)
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect00 (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x010:=(x01 x3):(T Xx)
% Found (x01 x3) as proof of (T Xx)
% Found ((x0 Xx) x3) as proof of (T Xx)
% Found (fun (x4:(U Xx))=> ((x0 Xx) x3)) as proof of (T Xx)
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((U Xx)->(T Xx))
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (fun (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (T Xx)
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect00 (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x010:=(x01 x3):(T Xx)
% Found (x01 x3) as proof of (T Xx)
% Found ((x0 Xx) x3) as proof of (T Xx)
% Found (fun (x4:(U Xx))=> ((x0 Xx) x3)) as proof of (T Xx)
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((U Xx)->(T Xx))
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (fun (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (T Xx)
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect00 (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x000:=(x00 x4):(T Xx)
% Found (x00 x4) as proof of (T Xx)
% Found ((x0 Xx) x4) as proof of (T Xx)
% Found (fun (x5:(U Xx))=> ((x0 Xx) x4)) as proof of (T Xx)
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((U Xx)->(T Xx))
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (fun (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (T Xx)
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect00 (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (fun (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x000:=(x00 x4):(T Xx)
% Found (x00 x4) as proof of (T Xx)
% Found ((x0 Xx) x4) as proof of (T Xx)
% Found (fun (x5:(U Xx))=> ((x0 Xx) x4)) as proof of (T Xx)
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((U Xx)->(T Xx))
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (fun (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (T Xx)
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect00 (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (fun (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x000:=(x00 x4):(T Xx)
% Found (x00 x4) as proof of (T Xx)
% Found ((x0 Xx) x4) as proof of (T Xx)
% Found (fun (x5:(U Xx))=> ((x0 Xx) x4)) as proof of (T Xx)
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((U Xx)->(T Xx))
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (fun (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (T Xx)
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect00 (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (fun (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found x010:=(x01 x3):(T Xx)
% Found (x01 x3) as proof of (T Xx)
% Found ((x0 Xx) x3) as proof of (T Xx)
% Found (fun (x4:(U Xx))=> ((x0 Xx) x3)) as proof of (T Xx)
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((U Xx)->(T Xx))
% Found (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))) as proof of (T Xx)
% Found (fun (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (T Xx)
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect00 (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x1:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x1) x00)) (T Xx)) (fun (x1:((and (S Xx)) (U Xx))) (x2:(not (T Xx)))=> (((fun (P:Type) (x3:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x3) x1)) (T Xx)) (fun (x3:(S Xx)) (x4:(U Xx))=> ((x0 Xx) x3)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x5):(T Xx)
% Found (x20 x5) as proof of (T Xx)
% Found ((x2 Xx) x5) as proof of (T Xx)
% Found (fun (x6:(U Xx))=> ((x2 Xx) x5)) as proof of (T Xx)
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((U Xx)->(T Xx))
% Found (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))) as proof of (T Xx)
% Found (fun (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (T Xx)
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of ((not (T Xx))->(T Xx))
% Found (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))) as proof of (((and (S Xx)) (U Xx))->((not (T Xx))->(T Xx)))
% Found (and_rect10 (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5))))) as proof of (T Xx)
% Found (fun (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x00:((and ((and (S Xx)) (U Xx))) (not (T Xx))))=> (((fun (P:Type) (x3:(((and (S Xx)) (U Xx))->((not (T Xx))->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) (not (T Xx))) P) x3) x00)) (T Xx)) (fun (x3:((and (S Xx)) (U Xx))) (x4:(not (T Xx)))=> (((fun (P:Type) (x5:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x5) x3)) (T Xx)) (fun (x5:(S Xx)) (x6:(U Xx))=> ((x2 Xx) x5)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) (not (T Xx)))->(T Xx))))
% Found x00:(S Xx)
% Found x00 as proof of (S Xx)
% Found x010:=(x01 x00):(U Xx)
% Found (x01 x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((x0 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x0 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found x100:=(x10 x00):(U Xx)
% Found (x10 x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((x1 Xx) x00) as proof of (U Xx)
% Found ((conj20 x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x1:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x1) x)) ((and (S Xx)) (U Xx))) (fun (x1:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x2:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x00) ((x1 Xx) x00)))) as proof of ((and (S Xx)) (U Xx))
% Found x000:=(x00 x4):(T Xx)
% Found (x00 x4) as proof of (T Xx)
% Found ((x0 Xx) x4) as proof of (T Xx)
% Found (fun (x5:(U Xx))=> ((x0 Xx) x4)) as proof of (T Xx)
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((U Xx)->(T Xx))
% Found (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect10 (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))) as proof of (T Xx)
% Found (fun (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (T Xx)
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect00 (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found ((and_rect0 (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4))))) as proof of (T Xx)
% Found (fun (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x0:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x1:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x2:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x2) x1)) (T Xx)) (fun (x2:((and (S Xx)) (U Xx))) (x3:((T Xx)->False))=> (((fun (P:Type) (x4:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x4) x2)) (T Xx)) (fun (x4:(S Xx)) (x5:(U Xx))=> ((x0 Xx) x4)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x1:(S Xx)
% Found x1 as proof of (S Xx)
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x6):(T Xx)
% Found (x20 x6) as proof of (T Xx)
% Found ((x2 Xx) x6) as proof of (T Xx)
% Found (fun (x7:(U Xx))=> ((x2 Xx) x6)) as proof of (T Xx)
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((U Xx)->(T Xx))
% Found (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)) as proof of ((S Xx)->((U Xx)->(T Xx)))
% Found (and_rect20 (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found ((and_rect2 (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))) as proof of (T Xx)
% Found (fun (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (T Xx)
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((T Xx)->False)->(T Xx))
% Found (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))) as proof of (((and (S Xx)) (U Xx))->(((T Xx)->False)->(T Xx)))
% Found (and_rect10 (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found ((and_rect1 (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6))))) as proof of (T Xx)
% Found (fun (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (T Xx)
% Found (fun (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of (forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx)))
% Found (fun (x2:(forall (Xx:a), ((S Xx)->(T Xx)))) (Xx:a) (x3:((and ((and (S Xx)) (U Xx))) ((T Xx)->False)))=> (((fun (P:Type) (x4:(((and (S Xx)) (U Xx))->(((T Xx)->False)->P)))=> (((((and_rect ((and (S Xx)) (U Xx))) ((T Xx)->False)) P) x4) x3)) (T Xx)) (fun (x4:((and (S Xx)) (U Xx))) (x5:((T Xx)->False))=> (((fun (P:Type) (x6:((S Xx)->((U Xx)->P)))=> (((((and_rect (S Xx)) (U Xx)) P) x6) x4)) (T Xx)) (fun (x6:(S Xx)) (x7:(U Xx))=> ((x2 Xx) x6)))))) as proof of ((forall (Xx:a), ((S Xx)->(T Xx)))->(forall (Xx:a), (((and ((and (S Xx)) (U Xx))) ((T Xx)->False))->(T Xx))))
% Found x3:(S Xx)
% Found x3 as proof of (S Xx)
% Found x000:=(x00 x3):(U Xx)
% Found (x00 x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((x0 Xx) x3) as proof of (U Xx)
% Found ((conj20 x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x3) ((x0 Xx) x3)) as proof of ((and (S Xx)) (U Xx))
% Found x200:=(x20 x1):(U Xx)
% Found (x20 x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((x2 Xx) x1) as proof of (U Xx)
% Found ((conj20 x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (((conj2 (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((and (S Xx)) (U Xx))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx)))
% Found (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1))) as proof of ((forall (Xx0:a), ((S Xx0)->(U Xx0)))->((forall (Xx0:a), ((T Xx0)->(U Xx0)))->((and (S Xx)) (U Xx))))
% Found (and_rect00 (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found ((and_rect0 ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% Found (((fun (P:Type) (x2:((forall (Xx:a), ((S Xx)->(U Xx)))->((forall (Xx:a), ((T Xx)->(U Xx)))->P)))=> (((((and_rect (forall (Xx:a), ((S Xx)->(U Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) P) x2) x)) ((and (S Xx)) (U Xx))) (fun (x2:(forall (Xx0:a), ((S Xx0)->(U Xx0)))) (x3:(forall (Xx0:a), ((T Xx0)->(U Xx0))))=> ((((conj (S Xx)) (U Xx)) x1) ((x2 Xx) x1)))) as proof of ((and (S Xx)) (U Xx))
% % SZS status GaveUp for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------