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
( P . k = [((P . k) `1),((P . k) `2)] & P . (k + 1) = [((P . (k + 1)) `1),((P . (k + 1)) `2)] )
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
( P . k = [((P . k) `1),((P . k) `2)] & P . (k + 1) = [((P . (k + 1)) `1),((P . (k + 1)) `2)] )
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
( P . k = [((P . k) `1),((P . k) `2)] & P . (k + 1) = [((P . (k + 1)) `1),((P . (k + 1)) `2)] )
let P be RedSequence of ==>.-relation TS; :: thesis: for k being Nat st k in dom P & k + 1 in dom P holds
( P . k = [((P . k) `1),((P . k) `2)] & P . (k + 1) = [((P . (k + 1)) `1),((P . (k + 1)) `2)] )
let k be Nat; :: thesis: ( k in dom P & k + 1 in dom P implies ( P . k = [((P . k) `1),((P . k) `2)] & P . (k + 1) = [((P . (k + 1)) `1),((P . (k + 1)) `2)] ) )
assume ( k in dom P & k + 1 in dom P ) ; :: thesis: ( P . k = [((P . k) `1),((P . k) `2)] & P . (k + 1) = [((P . (k + 1)) `1),((P . (k + 1)) `2)] )
then 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 Th47;
hence ( P . k = [((P . k) `1),((P . k) `2)] & P . (k + 1) = [((P . (k + 1)) `1),((P . (k + 1)) `2)] ) ; :: thesis: verum