let E be non empty set ; :: 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 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 of 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 of 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 A: ( 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 )] )
consider s being Element of TS, v being Element of E ^omega , t being Element of TS, w being Element of E ^omega such that
B: ( P . k = [s,v] & P . (k + 1) = [t,w] ) by A, ThRedSeq4;
thus ( P . k = [((P . k) `1 ),((P . k) `2 )] & P . (k + 1) = [((P . (k + 1)) `1 ),((P . (k + 1)) `2 )] ) by B, MCART_1:8; :: thesis: verum