TSTP Solution File: SEV026^5 by cocATP---0.2.0

View Problem - Process Solution 
%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV026^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n179.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 13:33:36 EDT 2014

% Result   : Timeout 300.04s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV026^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n179.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 07:36:06 CDT 2014
% % CPUTime  : 300.04 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (forall (Xr:(fofType->(fofType->Prop))), ((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and ((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:fofType), ((Xs Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xs Xx) Xy)) ((Xs Xy) Xz))->((Xs Xx) Xz))))) (forall (Xt:(fofType->(fofType->Prop))), (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))->(forall (Xa:fofType) (Xb:fofType), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))) of role conjecture named cTHM601_pme
% Conjecture to prove = (forall (Xr:(fofType->(fofType->Prop))), ((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and ((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:fofType), ((Xs Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xs Xx) Xy)) ((Xs Xy) Xz))->((Xs Xx) Xz))))) (forall (Xt:(fofType->(fofType->Prop))), (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))->(forall (Xa:fofType) (Xb:fofType), (((Xs Xa) Xb)->((Xt Xa) Xb))))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (Xr:(fofType->(fofType->Prop))), ((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and ((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:fofType), ((Xs Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xs Xx) Xy)) ((Xs Xy) Xz))->((Xs Xx) Xz))))) (forall (Xt:(fofType->(fofType->Prop))), (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))->(forall (Xa:fofType) (Xb:fofType), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))))']
% Parameter fofType:Type.
% Trying to prove (forall (Xr:(fofType->(fofType->Prop))), ((ex (fofType->(fofType->Prop))) (fun (Xs:(fofType->(fofType->Prop)))=> ((and ((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xs Xa) Xb)))) (forall (Xx:fofType), ((Xs Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xs Xx) Xy)->((Xs Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xs Xx) Xy)) ((Xs Xy) Xz))->((Xs Xx) Xz))))) (forall (Xt:(fofType->(fofType->Prop))), (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))->(forall (Xa:fofType) (Xb:fofType), (((Xs Xa) Xb)->((Xt Xa) Xb)))))))))
% Found x00:((x Xa) Xb)
% Instantiate: x:=Xt:(fofType->(fofType->Prop))
% Found x00 as proof of ((Xt Xa) Xb)
% Found (fun (x00:((x Xa) Xb))=> x00) as proof of ((Xt Xa) Xb)
% Found (fun (Xb:fofType) (x00:((x Xa) Xb))=> x00) as proof of (((x Xa) Xb)->((Xt Xa) Xb))
% Found (fun (Xa:fofType) (Xb:fofType) (x00:((x Xa) Xb))=> x00) as proof of (forall (Xb:fofType), (((x Xa) Xb)->((Xt Xa) Xb)))
% Found (fun (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))) (Xa:fofType) (Xb:fofType) (x00:((x Xa) Xb))=> x00) as proof of (forall (Xa:fofType) (Xb:fofType), (((x Xa) Xb)->((Xt Xa) Xb)))
% Found x0000:=(x000 x0):((Xt Xa) Xb)
% Found (x000 x0) as proof of ((Xt Xa) Xb)
% Found ((x00 Xt) x0) as proof of ((Xt Xa) Xb)
% Found ((x00 Xt) x0) as proof of ((Xt Xa) Xb)
% Found (fun (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of ((Xt Xa) Xb)
% Found (fun (Xb:fofType) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (((x Xa) Xb)->((Xt Xa) Xb))
% Found (fun (Xa:fofType) (Xb:fofType) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (forall (Xb:fofType), (((x Xa) Xb)->((Xt Xa) Xb)))
% Found (fun (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))) (Xa:fofType) (Xb:fofType) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (forall (Xa:fofType) (Xb:fofType), (((x Xa) Xb)->((Xt Xa) Xb)))
% Found (fun (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))) (Xa:fofType) (Xb:fofType) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))->(forall (Xa:fofType) (Xb:fofType), (((x Xa) Xb)->((Xt Xa) Xb))))
% Found (fun (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))) (Xa:fofType) (Xb:fofType) (x00:((x Xa) Xb))=> ((x00 Xt) x0)) as proof of (forall (Xt:(fofType->(fofType->Prop))), (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))->(forall (Xa:fofType) (Xb:fofType), (((x Xa) Xb)->((Xt Xa) Xb)))))
% Found x0:((Xr Xa) Xb)
% Instantiate: x:=Xr:(fofType->(fofType->Prop))
% Found (fun (x0:((Xr Xa) Xb))=> x0) as proof of ((x Xa) Xb)
% Found (fun (Xb:fofType) (x0:((Xr Xa) Xb))=> x0) as proof of (((Xr Xa) Xb)->((x Xa) Xb))
% Found (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb))=> x0) as proof of (forall (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))
% Found (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb))=> x0) as proof of (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))
% Found x03:((x Xx0) Xz)
% Found (fun (x04:((x Xy0) Xz))=> x03) as proof of ((x Xx0) Xz)
% Found (fun (x03:((x Xx0) Xz)) (x04:((x Xy0) Xz))=> x03) as proof of (((x Xy0) Xz)->((x Xx0) Xz))
% Found (fun (x03:((x Xx0) Xz)) (x04:((x Xy0) Xz))=> x03) as proof of (((x Xx0) Xz)->(((x Xy0) Xz)->((x Xx0) Xz)))
% Found (and_rect10 (fun (x03:((x Xx0) Xz)) (x04:((x Xy0) Xz))=> x03)) as proof of ((x Xx0) Xz)
% Found ((and_rect1 ((x Xx0) Xz)) (fun (x03:((x Xx0) Xz)) (x04:((x Xy0) Xz))=> x03)) as proof of ((x Xx0) Xz)
% Found (((fun (P:Type) (x1:(((x Xx0) Xz)->(((x Xy0) Xz)->P)))=> (((((and_rect ((x Xx0) Xz)) ((x Xy0) Xz)) P) x1) x02)) ((x Xx0) Xz)) (fun (x03:((x Xx0) Xz)) (x04:((x Xy0) Xz))=> x03)) as proof of ((x Xx0) Xz)
% Found (fun (x02:((and ((x Xx0) Xz)) ((x Xy0) Xz)))=> (((fun (P:Type) (x1:(((x Xx0) Xz)->(((x Xy0) Xz)->P)))=> (((((and_rect ((x Xx0) Xz)) ((x Xy0) Xz)) P) x1) x02)) ((x Xx0) Xz)) (fun (x03:((x Xx0) Xz)) (x04:((x Xy0) Xz))=> x03))) as proof of ((x Xx0) Xz)
% Found (fun (Xz0:fofType) (x02:((and ((x Xx0) Xz)) ((x Xy0) Xz)))=> (((fun (P:Type) (x1:(((x Xx0) Xz)->(((x Xy0) Xz)->P)))=> (((((and_rect ((x Xx0) Xz)) ((x Xy0) Xz)) P) x1) x02)) ((x Xx0) Xz)) (fun (x03:((x Xx0) Xz)) (x04:((x Xy0) Xz))=> x03))) as proof of (((and ((x Xx0) Xz)) ((x Xy0) Xz))->((x Xx0) Xz))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x02:((and ((x Xx0) Xz)) ((x Xy0) Xz)))=> (((fun (P:Type) (x1:(((x Xx0) Xz)->(((x Xy0) Xz)->P)))=> (((((and_rect ((x Xx0) Xz)) ((x Xy0) Xz)) P) x1) x02)) ((x Xx0) Xz)) (fun (x03:((x Xx0) Xz)) (x04:((x Xy0) Xz))=> x03))) as proof of (fofType->(((and ((x Xx0) Xz)) ((x Xy0) Xz))->((x Xx0) Xz)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x02:((and ((x Xx0) Xz)) ((x Xy0) Xz)))=> (((fun (P:Type) (x1:(((x Xx0) Xz)->(((x Xy0) Xz)->P)))=> (((((and_rect ((x Xx0) Xz)) ((x Xy0) Xz)) P) x1) x02)) ((x Xx0) Xz)) (fun (x03:((x Xx0) Xz)) (x04:((x Xy0) Xz))=> x03))) as proof of (forall (Xy0:fofType), (fofType->(((and ((x Xx0) Xz)) ((x Xy0) Xz))->((x Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x02:((and ((x Xx0) Xz)) ((x Xy0) Xz)))=> (((fun (P:Type) (x1:(((x Xx0) Xz)->(((x Xy0) Xz)->P)))=> (((((and_rect ((x Xx0) Xz)) ((x Xy0) Xz)) P) x1) x02)) ((x Xx0) Xz)) (fun (x03:((x Xx0) Xz)) (x04:((x Xy0) Xz))=> x03))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and ((x Xx0) Xz)) ((x Xy0) Xz))->((x Xx0) Xz))))
% Found x04:((x Xx) Xz0)
% Found (fun (x04:((x Xx) Xz0))=> x04) as proof of ((x Xx) Xz0)
% Found (fun (x03:((x Xx) Xy0)) (x04:((x Xx) Xz0))=> x04) as proof of (((x Xx) Xz0)->((x Xx) Xz0))
% Found (fun (x03:((x Xx) Xy0)) (x04:((x Xx) Xz0))=> x04) as proof of (((x Xx) Xy0)->(((x Xx) Xz0)->((x Xx) Xz0)))
% Found (and_rect10 (fun (x03:((x Xx) Xy0)) (x04:((x Xx) Xz0))=> x04)) as proof of ((x Xx) Xz0)
% Found ((and_rect1 ((x Xx) Xz0)) (fun (x03:((x Xx) Xy0)) (x04:((x Xx) Xz0))=> x04)) as proof of ((x Xx) Xz0)
% Found (((fun (P:Type) (x1:(((x Xx) Xy0)->(((x Xx) Xz0)->P)))=> (((((and_rect ((x Xx) Xy0)) ((x Xx) Xz0)) P) x1) x02)) ((x Xx) Xz0)) (fun (x03:((x Xx) Xy0)) (x04:((x Xx) Xz0))=> x04)) as proof of ((x Xx) Xz0)
% Found (fun (x02:((and ((x Xx) Xy0)) ((x Xx) Xz0)))=> (((fun (P:Type) (x1:(((x Xx) Xy0)->(((x Xx) Xz0)->P)))=> (((((and_rect ((x Xx) Xy0)) ((x Xx) Xz0)) P) x1) x02)) ((x Xx) Xz0)) (fun (x03:((x Xx) Xy0)) (x04:((x Xx) Xz0))=> x04))) as proof of ((x Xx) Xz0)
% Found (fun (Xz0:fofType) (x02:((and ((x Xx) Xy0)) ((x Xx) Xz0)))=> (((fun (P:Type) (x1:(((x Xx) Xy0)->(((x Xx) Xz0)->P)))=> (((((and_rect ((x Xx) Xy0)) ((x Xx) Xz0)) P) x1) x02)) ((x Xx) Xz0)) (fun (x03:((x Xx) Xy0)) (x04:((x Xx) Xz0))=> x04))) as proof of (((and ((x Xx) Xy0)) ((x Xx) Xz0))->((x Xx) Xz0))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x02:((and ((x Xx) Xy0)) ((x Xx) Xz0)))=> (((fun (P:Type) (x1:(((x Xx) Xy0)->(((x Xx) Xz0)->P)))=> (((((and_rect ((x Xx) Xy0)) ((x Xx) Xz0)) P) x1) x02)) ((x Xx) Xz0)) (fun (x03:((x Xx) Xy0)) (x04:((x Xx) Xz0))=> x04))) as proof of (forall (Xz0:fofType), (((and ((x Xx) Xy0)) ((x Xx) Xz0))->((x Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x02:((and ((x Xx) Xy0)) ((x Xx) Xz0)))=> (((fun (P:Type) (x1:(((x Xx) Xy0)->(((x Xx) Xz0)->P)))=> (((((and_rect ((x Xx) Xy0)) ((x Xx) Xz0)) P) x1) x02)) ((x Xx) Xz0)) (fun (x03:((x Xx) Xy0)) (x04:((x Xx) Xz0))=> x04))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and ((x Xx) Xy0)) ((x Xx) Xz0))->((x Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x02:((and ((x Xx) Xy0)) ((x Xx) Xz0)))=> (((fun (P:Type) (x1:(((x Xx) Xy0)->(((x Xx) Xz0)->P)))=> (((((and_rect ((x Xx) Xy0)) ((x Xx) Xz0)) P) x1) x02)) ((x Xx) Xz0)) (fun (x03:((x Xx) Xy0)) (x04:((x Xx) Xz0))=> x04))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and ((x Xx) Xy0)) ((x Xx) Xz0))->((x Xx) Xz0))))
% Found x1:((Xt Xx0) Xz)
% Found (fun (x2:((Xt Xy0) Xz))=> x1) as proof of ((Xt Xx0) Xz)
% Found (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1) as proof of (((Xt Xy0) Xz)->((Xt Xx0) Xz))
% Found (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1) as proof of (((Xt Xx0) Xz)->(((Xt Xy0) Xz)->((Xt Xx0) Xz)))
% Found (and_rect10 (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1)) as proof of ((Xt Xx0) Xz)
% Found ((and_rect1 ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1)) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1)) as proof of ((Xt Xx0) Xz)
% Found (fun (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1))) as proof of ((Xt Xx0) Xz)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1))) as proof of (((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1))) as proof of (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1))) as proof of (forall (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found x2:((Xt Xx) Xz0)
% Found (fun (x2:((Xt Xx) Xz0))=> x2) as proof of ((Xt Xx) Xz0)
% Found (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2) as proof of (((Xt Xx) Xz0)->((Xt Xx) Xz0))
% Found (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2) as proof of (((Xt Xx) Xy0)->(((Xt Xx) Xz0)->((Xt Xx) Xz0)))
% Found (and_rect10 (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2)) as proof of ((Xt Xx) Xz0)
% Found ((and_rect1 ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2)) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2)) as proof of ((Xt Xx) Xz0)
% Found (fun (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2))) as proof of ((Xt Xx) Xz0)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2))) as proof of (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2))) as proof of (forall (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))))
% Found x1:((Xt Xx0) Xz)
% Found (fun (x2:((Xt Xy0) Xz))=> x1) as proof of ((Xt Xx0) Xz)
% Found (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1) as proof of (((Xt Xy0) Xz)->((Xt Xx0) Xz))
% Found (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1) as proof of (((Xt Xx0) Xz)->(((Xt Xy0) Xz)->((Xt Xx0) Xz)))
% Found (and_rect10 (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1)) as proof of ((Xt Xx0) Xz)
% Found ((and_rect1 ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1)) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1)) as proof of ((Xt Xx0) Xz)
% Found (fun (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1))) as proof of ((Xt Xx0) Xz)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1))) as proof of (((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1))) as proof of (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1))) as proof of (forall (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x1:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:((Xt Xx0) Xz)) (x2:((Xt Xy0) Xz))=> x1))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found x2:((Xt Xx) Xz0)
% Found (fun (x2:((Xt Xx) Xz0))=> x2) as proof of ((Xt Xx) Xz0)
% Found (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2) as proof of (((Xt Xx) Xz0)->((Xt Xx) Xz0))
% Found (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2) as proof of (((Xt Xx) Xy0)->(((Xt Xx) Xz0)->((Xt Xx) Xz0)))
% Found (and_rect10 (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2)) as proof of ((Xt Xx) Xz0)
% Found ((and_rect1 ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2)) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2)) as proof of ((Xt Xx) Xz0)
% Found (fun (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2))) as proof of ((Xt Xx) Xz0)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2))) as proof of (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2))) as proof of (forall (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x1:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:((Xt Xx) Xy0)) (x2:((Xt Xx) Xz0))=> x2))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))))
% Found x10:=(x1 x020):((Xt Xx0) Xz)
% Found (x1 x020) as proof of ((Xt Xx0) Xz)
% Found (fun (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020)) as proof of ((Xt Xx0) Xz)
% Found (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020)) as proof of ((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))->((Xt Xx0) Xz))
% Found (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020)) as proof of ((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))->((Xt Xx0) Xz)))
% Found (and_rect10 (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020))) as proof of ((Xt Xx0) Xz)
% Found ((and_rect1 ((Xt Xx0) Xz)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020))) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))) P) x1) x02)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020))) as proof of ((Xt Xx0) Xz)
% Found (fun (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))) P) x1) x02)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020)))) as proof of ((Xt Xx0) Xz)
% Found (fun (x02:((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xy0) Xz)))) (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))) P) x1) x02)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020)))) as proof of (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx0) Xz))
% Found (fun (Xz0:fofType) (x02:((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xy0) Xz)))) (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))) P) x1) x02)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020)))) as proof of (((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xy0) Xz)))->(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx0) Xz)))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x02:((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xy0) Xz)))) (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))) P) x1) x02)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020)))) as proof of (fofType->(((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xy0) Xz)))->(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x02:((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xy0) Xz)))) (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))) P) x1) x02)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020)))) as proof of (forall (Xy0:fofType), (fofType->(((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xy0) Xz)))->(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx0) Xz)))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x02:((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xy0) Xz)))) (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz))) P) x1) x02)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xy0) Xz)))=> (x1 x020)))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx0) Xz))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xy0) Xz)))->(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx0) Xz)))))
% Found x20:=(x2 x020):((Xt Xx) Xz0)
% Found (x2 x020) as proof of ((Xt Xx) Xz0)
% Found (fun (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020)) as proof of ((Xt Xx) Xz0)
% Found (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020)) as proof of ((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))->((Xt Xx) Xz0))
% Found (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020)) as proof of ((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (and_rect10 (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020))) as proof of ((Xt Xx) Xz0)
% Found ((and_rect1 ((Xt Xx) Xz0)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020))) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))) P) x1) x02)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020))) as proof of ((Xt Xx) Xz0)
% Found (fun (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))) P) x1) x02)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020)))) as proof of ((Xt Xx) Xz0)
% Found (fun (x02:((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))) (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))) P) x1) x02)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020)))) as proof of (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xz0))
% Found (fun (Xz0:fofType) (x02:((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))) (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))) P) x1) x02)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020)))) as proof of (((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))->(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xz0)))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x02:((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))) (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))) P) x1) x02)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020)))) as proof of (forall (Xz0:fofType), (((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))->(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xz0))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x02:((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))) (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))) P) x1) x02)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020)))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))->(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xz0))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x02:((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))) (x020:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))))=> (((fun (P:Type) (x1:((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))->((((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))->P)))=> (((((and_rect (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0))) P) x1) x02)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xy0))) (x2:(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))=> (x2 x020)))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))) (forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0))))->((Xt Xx) Xy0))) (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))) (forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00))))->((Xt Xx) Xz0)))->(((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))->((Xt Xx) Xz0)))))
% Found x010:=(x01 x00):((Xt Xx) Xy)
% Found (x01 x00) as proof of ((Xt Xx) Xy)
% Found ((x0 Xt) x00) as proof of ((Xt Xx) Xy)
% Found ((x0 Xt) x00) as proof of ((Xt Xx) Xy)
% Found (x400 ((x0 Xt) x00)) as proof of ((Xt Xy) Xx)
% Found ((x40 Xy) ((x0 Xt) x00)) as proof of ((Xt Xy) Xx)
% Found (((x4 Xx) Xy) ((x0 Xt) x00)) as proof of ((Xt Xy) Xx)
% Found (fun (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))) as proof of ((Xt Xy) Xx)
% Found (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))) as proof of ((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->((Xt Xy) Xx))
% Found (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))) as proof of (((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->((Xt Xy) Xx)))
% Found (and_rect10 (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00)))) as proof of ((Xt Xy) Xx)
% Found ((and_rect1 ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00)))) as proof of ((Xt Xy) Xx)
% Found (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00)))) as proof of ((Xt Xy) Xx)
% Found (fun (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))))) as proof of ((Xt Xy) Xx)
% Found (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))))) as proof of ((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->((Xt Xy) Xx))
% Found (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))))) as proof of (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->((Xt Xy) Xx)))
% Found (and_rect00 (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00)))))) as proof of ((Xt Xy) Xx)
% Found ((and_rect0 ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00)))))) as proof of ((Xt Xy) Xx)
% Found (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00)))))) as proof of ((Xt Xy) Xx)
% Found (fun (x00:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))))))) as proof of ((Xt Xy) Xx)
% Found (fun (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))))))) as proof of (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))->((Xt Xy) Xx))
% Found (fun (x0:((x Xx) Xy)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))))))) as proof of ((x Xy) Xx)
% Found (fun (Xy:fofType) (x0:((x Xx) Xy)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))))))) as proof of (((x Xx) Xy)->((x Xy) Xx))
% Found (fun (Xx:fofType) (Xy:fofType) (x0:((x Xx) Xy)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))))))) as proof of (forall (Xy:fofType), (((x Xx) Xy)->((x Xy) Xx)))
% Found (fun (Xx:fofType) (Xy:fofType) (x0:((x Xx) Xy)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00))))))) as proof of (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x Xy) Xx)))
% Found x60:=(x6 Xx):((Xt Xx) Xx)
% Found (x6 Xx) as proof of ((Xt Xx) Xx)
% Found (fun (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)) as proof of ((Xt Xx) Xx)
% Found (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)) as proof of ((forall (Xx0:fofType), ((Xt Xx0) Xx0))->((Xt Xx) Xx))
% Found (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)) as proof of ((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->((Xt Xx) Xx)))
% Found (and_rect20 (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))) as proof of ((Xt Xx) Xx)
% Found ((and_rect2 ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))) as proof of ((Xt Xx) Xx)
% Found (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))) as proof of ((Xt Xx) Xx)
% Found (fun (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))) as proof of ((Xt Xx) Xx)
% Found (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))) as proof of ((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->((Xt Xx) Xx))
% Found (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))) as proof of (((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->((Xt Xx) Xx)))
% Found (and_rect10 (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))))) as proof of ((Xt Xx) Xx)
% Found ((and_rect1 ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))))) as proof of ((Xt Xx) Xx)
% Found (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))))) as proof of ((Xt Xx) Xx)
% Found (fun (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))) as proof of ((Xt Xx) Xx)
% Found (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))) as proof of ((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->((Xt Xx) Xx))
% Found (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))) as proof of (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->((Xt Xx) Xx)))
% Found (and_rect00 (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))))))) as proof of ((Xt Xx) Xx)
% Found ((and_rect0 ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))))))) as proof of ((Xt Xx) Xx)
% Found (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))))))) as proof of ((Xt Xx) Xx)
% Found (fun (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))))) as proof of ((Xt Xx) Xx)
% Found (fun (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))))) as proof of (((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))->((Xt Xx) Xx))
% Found (fun (Xx:fofType) (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))))) as proof of ((x Xx) Xx)
% Found (fun (Xx:fofType) (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))))) as proof of (forall (Xx:fofType), ((x Xx) Xx))
% Found x5000:=(x500 x0):((Xt Xa) Xb)
% Found (x500 x0) as proof of ((Xt Xa) Xb)
% Found ((x50 Xb) x0) as proof of ((Xt Xa) Xb)
% Found (((x5 Xa) Xb) x0) as proof of ((Xt Xa) Xb)
% Found (fun (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)) as proof of ((Xt Xa) Xb)
% Found (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)) as proof of ((forall (Xx:fofType), ((Xt Xx) Xx))->((Xt Xa) Xb))
% Found (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)) as proof of ((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->((Xt Xa) Xb)))
% Found (and_rect20 (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))) as proof of ((Xt Xa) Xb)
% Found ((and_rect2 ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))) as proof of ((Xt Xa) Xb)
% Found (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))) as proof of ((Xt Xa) Xb)
% Found (fun (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))) as proof of ((Xt Xa) Xb)
% Found (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->((Xt Xa) Xb))
% Found (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))) as proof of (((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->((Xt Xa) Xb)))
% Found (and_rect10 (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))) as proof of ((Xt Xa) Xb)
% Found ((and_rect1 ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))) as proof of ((Xt Xa) Xb)
% Found (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))) as proof of ((Xt Xa) Xb)
% Found (fun (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))))) as proof of ((Xt Xa) Xb)
% Found (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))))) as proof of ((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->((Xt Xa) Xb))
% Found (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))))) as proof of (((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->((Xt Xa) Xb)))
% Found (and_rect00 (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))) as proof of ((Xt Xa) Xb)
% Found ((and_rect0 ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))) as proof of ((Xt Xa) Xb)
% Found (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))) as proof of ((Xt Xa) Xb)
% Found (fun (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))))))) as proof of ((Xt Xa) Xb)
% Found (fun (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))))))) as proof of (((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))->((Xt Xa) Xb))
% Found (fun (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))))))) as proof of ((x Xa) Xb)
% Found (fun (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))))))) as proof of (((Xr Xa) Xb)->((x Xa) Xb))
% Found (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))))))) as proof of (forall (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))
% Found (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0)))))))) as proof of (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))
% Found ((conj30 (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))))) (fun (Xx:fofType) (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))))))))) as proof of ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx)))
% Found (((conj3 (forall (Xx:fofType), ((x Xx) Xx))) (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))))) (fun (Xx:fofType) (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))))))))) as proof of ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx)))
% Found ((((conj (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx))) (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))))) (fun (Xx:fofType) (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))))))))) as proof of ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx)))
% Found ((((conj (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx))) (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))))) (fun (Xx:fofType) (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx))))))))) as proof of ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx)))
% Found ((conj20 ((((conj (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx))) (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))))) (fun (Xx:fofType) (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))))))) (fun (Xx:fofType) (Xy:fofType) (x0:((x Xx) Xy)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00)))))))) as proof of ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x Xy) Xx))))
% Found (((conj2 (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x Xy) Xx)))) ((((conj (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx))) (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))))) (fun (Xx:fofType) (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))))))) (fun (Xx:fofType) (Xy:fofType) (x0:((x Xx) Xy)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00)))))))) as proof of ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x Xy) Xx))))
% Found ((((conj ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x Xy) Xx)))) ((((conj (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx))) (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))))) (fun (Xx:fofType) (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))))))) (fun (Xx:fofType) (Xy:fofType) (x0:((x Xx) Xy)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00)))))))) as proof of ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x Xy) Xx))))
% Found ((((conj ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x Xy) Xx)))) ((((conj (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx))) (fun (Xa:fofType) (Xb:fofType) (x0:((Xr Xa) Xb)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))->((forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz)))) P) x1) x00)) ((Xt Xa) Xb)) (fun (x1:((and ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))) (x2:(forall (Xx:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx) Xy)) ((Xt Xy) Xz))->((Xt Xx) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))->((forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))->P)))=> (((((and_rect ((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx)))) P) x3) x1)) ((Xt Xa) Xb)) (fun (x3:((and (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx)))) (x4:(forall (Xx:fofType) (Xy:fofType), (((Xt Xx) Xy)->((Xt Xy) Xx))))=> (((fun (P:Type) (x5:((forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))->((forall (Xx:fofType), ((Xt Xx) Xx))->P)))=> (((((and_rect (forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (forall (Xx:fofType), ((Xt Xx) Xx))) P) x5) x3)) ((Xt Xa) Xb)) (fun (x5:(forall (Xa0:fofType) (Xb0:fofType), (((Xr Xa0) Xb0)->((Xt Xa0) Xb0)))) (x6:(forall (Xx:fofType), ((Xt Xx) Xx)))=> (((x5 Xa) Xb) x0))))))))) (fun (Xx:fofType) (Xt:(fofType->(fofType->Prop))) (x0:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))->((forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz)))) P) x1) x0)) ((Xt Xx) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))) (x2:(forall (Xx0:fofType) (Xy:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy)) ((Xt Xy) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0)))) P) x3) x1)) ((Xt Xx) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy:fofType), (((Xt Xx0) Xy)->((Xt Xy) Xx0))))=> (((fun (P:Type) (x5:((forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))->((forall (Xx0:fofType), ((Xt Xx0) Xx0))->P)))=> (((((and_rect (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0))) P) x5) x3)) ((Xt Xx) Xx)) (fun (x5:(forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (x6:(forall (Xx0:fofType), ((Xt Xx0) Xx0)))=> (x6 Xx)))))))))) (fun (Xx:fofType) (Xy:fofType) (x0:((x Xx) Xy)) (Xt:(fofType->(fofType->Prop))) (x00:((and ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))))=> (((fun (P:Type) (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))->((forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))->P)))=> (((((and_rect ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))) P) x1) x00)) ((Xt Xy) Xx)) (fun (x1:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))) (x2:(forall (Xx0:fofType) (Xy0:fofType) (Xz:fofType), (((and ((Xt Xx0) Xy0)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))=> (((fun (P:Type) (x3:(((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))->((forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))->P)))=> (((((and_rect ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0)))) P) x3) x1)) ((Xt Xy) Xx)) (fun (x3:((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (x4:(forall (Xx0:fofType) (Xy0:fofType), (((Xt Xx0) Xy0)->((Xt Xy0) Xx0))))=> (((x4 Xx) Xy) ((x0 Xt) x00)))))))) as proof of ((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((x Xa) Xb)))) (forall (Xx:fofType), ((x Xx) Xx)))) (forall (Xx:fofType) (Xy:fofType), (((x Xx) Xy)->((x Xy) Xx))))
% Found x4:((Xt Xx) Xz0)
% Found (fun (x4:((Xt Xx) Xz0))=> x4) as proof of ((Xt Xx) Xz0)
% Found (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4) as proof of (((Xt Xx) Xz0)->((Xt Xx) Xz0))
% Found (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4) as proof of (((Xt Xx) Xy0)->(((Xt Xx) Xz0)->((Xt Xx) Xz0)))
% Found (and_rect20 (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found ((and_rect2 ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found (fun (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of ((Xt Xx) Xz0)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (forall (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))))
% Found x3:((Xt Xx0) Xz)
% Found (fun (x4:((Xt Xy0) Xz))=> x3) as proof of ((Xt Xx0) Xz)
% Found (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3) as proof of (((Xt Xy0) Xz)->((Xt Xx0) Xz))
% Found (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3) as proof of (((Xt Xx0) Xz)->(((Xt Xy0) Xz)->((Xt Xx0) Xz)))
% Found (and_rect20 (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found ((and_rect2 ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found (fun (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of ((Xt Xx0) Xz)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (forall (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found x3:((Xt Xx0) Xz)
% Found (fun (x4:((Xt Xy0) Xz))=> x3) as proof of ((Xt Xx0) Xz)
% Found (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3) as proof of (((Xt Xy0) Xz)->((Xt Xx0) Xz))
% Found (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3) as proof of (((Xt Xx0) Xz)->(((Xt Xy0) Xz)->((Xt Xx0) Xz)))
% Found (and_rect20 (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found ((and_rect2 ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found (fun (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of ((Xt Xx0) Xz)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (forall (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found x4:((Xt Xx) Xz0)
% Found (fun (x4:((Xt Xx) Xz0))=> x4) as proof of ((Xt Xx) Xz0)
% Found (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4) as proof of (((Xt Xx) Xz0)->((Xt Xx) Xz0))
% Found (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4) as proof of (((Xt Xx) Xy0)->(((Xt Xx) Xz0)->((Xt Xx) Xz0)))
% Found (and_rect20 (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found ((and_rect2 ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found (fun (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of ((Xt Xx) Xz0)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (forall (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))))
% Found x4:((Xt Xx) Xz0)
% Found (fun (x4:((Xt Xx) Xz0))=> x4) as proof of ((Xt Xx) Xz0)
% Found (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4) as proof of (((Xt Xx) Xz0)->((Xt Xx) Xz0))
% Found (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4) as proof of (((Xt Xx) Xy0)->(((Xt Xx) Xz0)->((Xt Xx) Xz0)))
% Found (and_rect20 (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found ((and_rect2 ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found (fun (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of ((Xt Xx) Xz0)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (forall (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))))
% Found x3:((Xt Xx0) Xz)
% Found (fun (x4:((Xt Xy0) Xz))=> x3) as proof of ((Xt Xx0) Xz)
% Found (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3) as proof of (((Xt Xy0) Xz)->((Xt Xx0) Xz))
% Found (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3) as proof of (((Xt Xx0) Xz)->(((Xt Xy0) Xz)->((Xt Xx0) Xz)))
% Found (and_rect20 (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found ((and_rect2 ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found (fun (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of ((Xt Xx0) Xz)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (forall (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found x3:((Xt Xx0) Xz)
% Found (fun (x4:((Xt Xy0) Xz))=> x3) as proof of ((Xt Xx0) Xz)
% Found (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3) as proof of (((Xt Xy0) Xz)->((Xt Xx0) Xz))
% Found (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3) as proof of (((Xt Xx0) Xz)->(((Xt Xy0) Xz)->((Xt Xx0) Xz)))
% Found (and_rect20 (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found ((and_rect2 ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found (fun (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of ((Xt Xx0) Xz)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (forall (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found x4:((Xt Xx) Xz0)
% Found (fun (x4:((Xt Xx) Xz0))=> x4) as proof of ((Xt Xx) Xz0)
% Found (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4) as proof of (((Xt Xx) Xz0)->((Xt Xx) Xz0))
% Found (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4) as proof of (((Xt Xx) Xy0)->(((Xt Xx) Xz0)->((Xt Xx) Xz0)))
% Found (and_rect20 (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found ((and_rect2 ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found (fun (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of ((Xt Xx) Xz0)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (forall (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))))
% Found x3:((Xt Xx0) Xz)
% Found (fun (x4:((Xt Xy0) Xz))=> x3) as proof of ((Xt Xx0) Xz)
% Found (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3) as proof of (((Xt Xy0) Xz)->((Xt Xx0) Xz))
% Found (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3) as proof of (((Xt Xx0) Xz)->(((Xt Xy0) Xz)->((Xt Xx0) Xz)))
% Found (and_rect20 (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found ((and_rect2 ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3)) as proof of ((Xt Xx0) Xz)
% Found (fun (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of ((Xt Xx0) Xz)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (forall (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz)))=> (((fun (P:Type) (x3:(((Xt Xx0) Xz)->(((Xt Xy0) Xz)->P)))=> (((((and_rect ((Xt Xx0) Xz)) ((Xt Xy0) Xz)) P) x3) x03)) ((Xt Xx0) Xz)) (fun (x3:((Xt Xx0) Xz)) (x4:((Xt Xy0) Xz))=> x3))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and ((Xt Xx0) Xz)) ((Xt Xy0) Xz))->((Xt Xx0) Xz))))
% Found x4:((Xt Xx) Xz0)
% Found (fun (x4:((Xt Xx) Xz0))=> x4) as proof of ((Xt Xx) Xz0)
% Found (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4) as proof of (((Xt Xx) Xz0)->((Xt Xx) Xz0))
% Found (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4) as proof of (((Xt Xx) Xy0)->(((Xt Xx) Xz0)->((Xt Xx) Xz0)))
% Found (and_rect20 (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found ((and_rect2 ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4)) as proof of ((Xt Xx) Xz0)
% Found (fun (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of ((Xt Xx) Xz0)
% Found (fun (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (forall (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0)))=> (((fun (P:Type) (x3:(((Xt Xx) Xy0)->(((Xt Xx) Xz0)->P)))=> (((((and_rect ((Xt Xx) Xy0)) ((Xt Xx) Xz0)) P) x3) x03)) ((Xt Xx) Xz0)) (fun (x3:((Xt Xx) Xy0)) (x4:((Xt Xx) Xz0))=> x4))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and ((Xt Xx) Xy0)) ((Xt Xx) Xz0))->((Xt Xx) Xz0))))
% Found x20:=(x2 x200):((Xt Xx0) Xz)
% Found (x2 x200) as proof of ((Xt Xx0) Xz)
% Found (fun (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)) as proof of ((Xt Xx0) Xz)
% Found (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)) as proof of (((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->((Xt Xx0) Xz))
% Found (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)) as proof of (((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->((Xt Xx0) Xz)))
% Found (and_rect20 (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200))) as proof of ((Xt Xx0) Xz)
% Found ((and_rect2 ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200))) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200))) as proof of ((Xt Xx0) Xz)
% Found (fun (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of ((Xt Xx0) Xz)
% Found (fun (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of ((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz))
% Found (fun (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of (((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz)))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of (fofType->(((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of (forall (Xy0:fofType), (fofType->(((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz)))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz)))))
% Found x30:=(x3 x200):((Xt Xx) Xz0)
% Found (x3 x200) as proof of ((Xt Xx) Xz0)
% Found (fun (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)) as proof of ((Xt Xx) Xz0)
% Found (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)) as proof of (((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->((Xt Xx) Xz0))
% Found (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)) as proof of (((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (and_rect20 (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200))) as proof of ((Xt Xx) Xz0)
% Found ((and_rect2 ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200))) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200))) as proof of ((Xt Xx) Xz0)
% Found (fun (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of ((Xt Xx) Xz0)
% Found (fun (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of ((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0))
% Found (fun (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of (((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0)))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of (forall (Xz0:fofType), (((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0)))))
% Found x20:=(x2 x200):((Xt Xx0) Xz)
% Found (x2 x200) as proof of ((Xt Xx0) Xz)
% Found (fun (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)) as proof of ((Xt Xx0) Xz)
% Found (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)) as proof of (((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->((Xt Xx0) Xz))
% Found (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)) as proof of (((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->((Xt Xx0) Xz)))
% Found (and_rect20 (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200))) as proof of ((Xt Xx0) Xz)
% Found ((and_rect2 ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200))) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200))) as proof of ((Xt Xx0) Xz)
% Found (fun (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of ((Xt Xx0) Xz)
% Found (fun (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of ((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz))
% Found (fun (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of (((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz)))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of (fofType->(((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of (forall (Xy0:fofType), (fofType->(((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz)))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))->(((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))) P) x2) x03)) ((Xt Xx0) Xz)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) (x3:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))=> (x2 x200)))) as proof of (forall (Xx0:fofType) (Xy0:fofType), (fofType->(((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz)))))
% Found x30:=(x3 x200):((Xt Xx) Xz0)
% Found (x3 x200) as proof of ((Xt Xx) Xz0)
% Found (fun (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)) as proof of ((Xt Xx) Xz0)
% Found (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)) as proof of (((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->((Xt Xx) Xz0))
% Found (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)) as proof of (((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->((Xt Xx) Xz0)))
% Found (and_rect20 (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200))) as proof of ((Xt Xx) Xz0)
% Found ((and_rect2 ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200))) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200))) as proof of ((Xt Xx) Xz0)
% Found (fun (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of ((Xt Xx) Xz0)
% Found (fun (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of ((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0))
% Found (fun (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of (((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0)))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of (forall (Xz0:fofType), (((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) (x200:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x2:(((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))->(((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))->P)))=> (((((and_rect ((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))) P) x2) x03)) ((Xt Xx) Xz0)) (fun (x2:((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0))) (x3:((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))=> (x3 x200)))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and ((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0))) ((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0)))))
% Found x210:=(x21 x20):((Xt Xx) Xz0)
% Found (x21 x20) as proof of ((Xt Xx) Xz0)
% Found ((x2 x100) x20) as proof of ((Xt Xx) Xz0)
% Found (fun (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20)) as proof of ((Xt Xx) Xz0)
% Found (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20)) as proof of ((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->((Xt Xx) Xz0))
% Found (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20)) as proof of ((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->((Xt Xx) Xz0)))
% Found (and_rect20 (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20))) as proof of ((Xt Xx) Xz0)
% Found ((and_rect2 ((Xt Xx) Xz0)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20))) as proof of ((Xt Xx) Xz0)
% Found (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20))) as proof of ((Xt Xx) Xz0)
% Found (fun (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20)))) as proof of ((Xt Xx) Xz0)
% Found (fun (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20)))) as proof of ((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0))
% Found (fun (x03:((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))) (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20)))) as proof of (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0)))
% Found (fun (Xz0:fofType) (x03:((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))) (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20)))) as proof of (((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))->(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0))))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))) (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20)))) as proof of (forall (Xz0:fofType), (((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))->(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0)))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))) (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20)))) as proof of (forall (Xy0:fofType) (Xz0:fofType), (((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))->(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0)))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))) (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0)))) P) x1) x03)) ((Xt Xx) Xz0)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xy0)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))=> ((x2 x100) x20)))) as proof of (fofType->(forall (Xy0:fofType) (Xz0:fofType), (((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx) Xy0)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xx) Xz0))))->(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx) Xz0))))))
% Found x101:=(x10 x20):((Xt Xx0) Xz)
% Found (x10 x20) as proof of ((Xt Xx0) Xz)
% Found ((x1 x100) x20) as proof of ((Xt Xx0) Xz)
% Found (fun (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20)) as proof of ((Xt Xx0) Xz)
% Found (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20)) as proof of ((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))->((Xt Xx0) Xz))
% Found (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20)) as proof of ((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))->((Xt Xx0) Xz)))
% Found (and_rect20 (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20))) as proof of ((Xt Xx0) Xz)
% Found ((and_rect2 ((Xt Xx0) Xz)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20))) as proof of ((Xt Xx0) Xz)
% Found (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20))) as proof of ((Xt Xx0) Xz)
% Found (fun (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20)))) as proof of ((Xt Xx0) Xz)
% Found (fun (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20)))) as proof of ((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz))
% Found (fun (x03:((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz))))) (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20)))) as proof of (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz)))
% Found (fun (Xz0:fofType) (x03:((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz))))) (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20)))) as proof of (((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz))))->(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz))))
% Found (fun (Xy0:fofType) (Xz0:fofType) (x03:((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz))))) (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20)))) as proof of (fofType->(((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz))))->(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz)))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz))))) (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz))))=> ((x1 x100) x20)))) as proof of (forall (Xy0:fofType), (fofType->(((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz))))->(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))->((forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00)))->((Xt Xx0) Xz))))))
% Found (fun (Xx0:fofType) (Xy0:fofType) (Xz0:fofType) (x03:((and (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx0:fofType), ((Xt Xx0) Xx0)))) (forall (Xx0:fofType) (Xy00:fofType), (((Xt Xx0) Xy00)->((Xt Xy00) Xx0))))->((forall (Xx0:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx0) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx0) Xz00)))->((Xt Xy0) Xz))))) (x100:((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy0:fofType), (((Xt Xx00) Xy0)->((Xt Xy0) Xx00))))) (x20:(forall (Xx00:fofType) (Xy0:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy0)) ((Xt Xy0) Xz00))->((Xt Xx00) Xz00))))=> (((fun (P:Type) (x1:((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))->((((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))->P)))=> (((((and_rect (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz00:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz00))->((Xt Xx00) Xz00)))->((Xt Xy0) Xz)))) P) x1) x03)) ((Xt Xx0) Xz)) (fun (x1:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->((Xt Xy00) Xx00))))->((forall (Xx00:fofType) (Xy00:fofType) (Xz0:fofType), (((and ((Xt Xx00) Xy00)) ((Xt Xy00) Xz0))->((Xt Xx00) Xz0)))->((Xt Xx0) Xz)))) (x2:(((and ((and (forall (Xa:fofType) (Xb:fofType), (((Xr Xa) Xb)->((Xt Xa) Xb)))) (forall (Xx00:fofType), ((Xt Xx00) Xx00)))) (forall (Xx00:fofType) (Xy00:fofType), (((Xt Xx00) Xy00)->
% EOF
%------------------------------------------------------------------------------