let s be non empty typealg ; :: thesis: for p being Proof of s
for v being Element of dom p st (p . v) `2 = 2 holds
ex w being Element of dom p ex T being FinSequence of the carrier of s ex x, y being type of s st
( w = v ^ <*0 *> & (p . v) `1 = [T,(y \ x)] & (p . w) `1 = [(<*y*> ^ T),x] )
let p be Proof of s; :: thesis: for v being Element of dom p st (p . v) `2 = 2 holds
ex w being Element of dom p ex T being FinSequence of the carrier of s ex x, y being type of s st
( w = v ^ <*0 *> & (p . v) `1 = [T,(y \ x)] & (p . w) `1 = [(<*y*> ^ T),x] )
let v be Element of dom p; :: thesis: ( (p . v) `2 = 2 implies ex w being Element of dom p ex T being FinSequence of the carrier of s ex x, y being type of s st
( w = v ^ <*0 *> & (p . v) `1 = [T,(y \ x)] & (p . w) `1 = [(<*y*> ^ T),x] ) )
A1: v is correct by Def13;
assume A2: (p . v) `2 = 2 ; :: thesis: ex w being Element of dom p ex T being FinSequence of the carrier of s ex x, y being type of s st
( w = v ^ <*0 *> & (p . v) `1 = [T,(y \ x)] & (p . w) `1 = [(<*y*> ^ T),x] )
then A3: ex T being FinSequence of the carrier of s ex x, y being type of s st
( (p . v) `1 = [T,(y \ x)] & (p . (v ^ <*0 *>)) `1 = [(<*y*> ^ T),x] ) by A1, Def5;
branchdeg v = 1 by A1, A2, Def5;
then v ^ <*0 *> in dom p by Th1;
hence ex w being Element of dom p ex T being FinSequence of the carrier of s ex x, y being type of s st
( w = v ^ <*0 *> & (p . v) `1 = [T,(y \ x)] & (p . w) `1 = [(<*y*> ^ T),x] ) by A3; :: thesis: verum