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 s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( P . k = [s,v] & P . (k + 1) = [t,w] )
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 s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( P . k = [s,v] & P . (k + 1) = [t,w] )
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 s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( P . k = [s,v] & P . (k + 1) = [t,w] )
let P be RedSequence of ==>.-relation TS; :: thesis: for k being Nat st k in dom P & k + 1 in dom P holds
ex s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( P . k = [s,v] & P . (k + 1) = [t,w] )
let k be Nat; :: thesis: ( k in dom P & k + 1 in dom P implies ex s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( P . k = [s,v] & P . (k + 1) = [t,w] ) )
assume ( k in dom P & k + 1 in dom P ) ; :: thesis: ex s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( P . k = [s,v] & P . (k + 1) = [t,w] )
then [(P . k),(P . (k + 1))] in ==>.-relation TS by REWRITE1:def 2;
hence ex s being Element of TS ex v being Element of E ^omega ex t being Element of TS ex w being Element of E ^omega st
( P . k = [s,v] & P . (k + 1) = [t,w] ) by Th31; :: thesis: verum