let E be non empty set ; :: thesis: for F being Subset of (E ^omega)
for TS being non empty transition-system over F
for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P holds
ex v, w being Element of E ^omega st
( v = (P . (k + 1)) `2 & (P . k) `1 ,w -->. (P . (k + 1)) `1 ,TS & (P . k) `2 = w ^ v )
let F be Subset of (E ^omega); :: thesis: for TS being non empty transition-system over F
for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P holds
ex v, w being Element of E ^omega st
( v = (P . (k + 1)) `2 & (P . k) `1 ,w -->. (P . (k + 1)) `1 ,TS & (P . k) `2 = w ^ v )
let TS be non empty transition-system over F; :: thesis: for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P holds
ex v, w being Element of E ^omega st
( v = (P . (k + 1)) `2 & (P . k) `1 ,w -->. (P . (k + 1)) `1 ,TS & (P . k) `2 = w ^ v )
let P be RedSequence of ==>.-relation TS; :: thesis: for k being Nat st k in dom P & k + 1 in dom P holds
ex v, w being Element of E ^omega st
( v = (P . (k + 1)) `2 & (P . k) `1 ,w -->. (P . (k + 1)) `1 ,TS & (P . k) `2 = w ^ v )
let k be Nat; :: thesis: ( k in dom P & k + 1 in dom P implies ex v, w being Element of E ^omega st
( v = (P . (k + 1)) `2 & (P . k) `1 ,w -->. (P . (k + 1)) `1 ,TS & (P . k) `2 = w ^ v ) )
assume A1: ( k in dom P & k + 1 in dom P ) ; :: thesis: ex v, w being Element of E ^omega st
( v = (P . (k + 1)) `2 & (P . k) `1 ,w -->. (P . (k + 1)) `1 ,TS & (P . k) `2 = w ^ v )
consider s being Element of TS, u being Element of E ^omega , t being Element of TS, v being Element of E ^omega such that
A2: P . k = [s,u] and
A3: P . (k + 1) = [t,v] by A1, Th47;
[[s,u],[t,v]] in ==>.-relation TS by A1, A2, A3, REWRITE1:def 2;
then consider v1, w1 being Element of E ^omega such that
A4: v1 = v and
A5: s,w1 -->. t,TS and
A6: u = w1 ^ v1 by Th35;
take v1 ; :: thesis: ex w being Element of E ^omega st
( v1 = (P . (k + 1)) `2 & (P . k) `1 ,w -->. (P . (k + 1)) `1 ,TS & (P . k) `2 = w ^ v1 )
take w1 ; :: thesis: ( v1 = (P . (k + 1)) `2 & (P . k) `1 ,w1 -->. (P . (k + 1)) `1 ,TS & (P . k) `2 = w1 ^ v1 )
thus v1 = (P . (k + 1)) `2 by A3, A4; :: thesis: ( (P . k) `1 ,w1 -->. (P . (k + 1)) `1 ,TS & (P . k) `2 = w1 ^ v1 )
(P . k) `1 ,w1 -->. t,TS by A2, A5;
hence ( (P . k) `1 ,w1 -->. (P . (k + 1)) `1 ,TS & (P . k) `2 = w1 ^ v1 ) by A2, A3, A6; :: thesis: verum