let x, y be set ; :: thesis: for E being non empty set
for u, v being Element of E ^omega
for F being Subset of (E ^omega )
for TS being non empty transition-system of F
for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P & P . k = [x,u] & P . (k + 1) = [y,v] holds
ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
let E be non empty set ; :: thesis: for u, v being Element of E ^omega
for F being Subset of (E ^omega )
for TS being non empty transition-system of F
for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P & P . k = [x,u] & P . (k + 1) = [y,v] holds
ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
let u, v be Element of E ^omega ; :: thesis: for F being Subset of (E ^omega )
for TS being non empty transition-system of F
for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P & P . k = [x,u] & P . (k + 1) = [y,v] holds
ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
let F be Subset of (E ^omega ); :: thesis: for TS being non empty transition-system of F
for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P & P . k = [x,u] & P . (k + 1) = [y,v] holds
ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
let TS be non empty transition-system of F; :: thesis: for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P & P . k = [x,u] & P . (k + 1) = [y,v] holds
ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
let P be RedSequence of ==>.-relation TS; :: thesis: for k being Nat st k in dom P & k + 1 in dom P & P . k = [x,u] & P . (k + 1) = [y,v] holds
ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
let k be Nat; :: thesis: ( k in dom P & k + 1 in dom P & P . k = [x,u] & P . (k + 1) = [y,v] implies ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v ) )
assume that
A1: ( k in dom P & k + 1 in dom P ) and
A2: ( P . k = [x,u] & P . (k + 1) = [y,v] ) ; :: thesis: ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
[[x,u],[y,v]] in ==>.-relation TS by A1, A2, REWRITE1:def 2;
hence ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v ) by ThRel40; :: thesis: verum