let E be non empty set ; :: thesis: for F being Subset of (E ^omega)
for TS being non empty transition-system over F st not <%> E in rng (dom the Tran of TS) holds
for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P holds
(P . k) `2 <> (P . (k + 1)) `2
let F be Subset of (E ^omega); :: thesis: for TS being non empty transition-system over F st not <%> E in rng (dom the Tran of TS) holds
for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P holds
(P . k) `2 <> (P . (k + 1)) `2
let TS be non empty transition-system over F; :: thesis: ( not <%> E in rng (dom the Tran of TS) implies for P being RedSequence of ==>.-relation TS
for k being Nat st k in dom P & k + 1 in dom P holds
(P . k) `2 <> (P . (k + 1)) `2 )
assume A1: not <%> E in rng (dom the Tran of TS) ; :: 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) `2 <> (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) `2 <> (P . (k + 1)) `2
let k be Nat; :: thesis: ( k in dom P & k + 1 in dom P implies (P . k) `2 <> (P . (k + 1)) `2 )
assume A2: ( k in dom P & k + 1 in dom P ) ; :: thesis: (P . k) `2 <> (P . (k + 1)) `2
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
A3: P . k = [s,u] and
A4: P . (k + 1) = [t,v] by A2, Th47;
[[s,u],[t,v]] in ==>.-relation TS by A2, A3, A4, REWRITE1:def 2;
then u <> v by A1, Th43;
then (P . k) `2 <> v by A3;
hence (P . k) `2 <> (P . (k + 1)) `2 by A4; :: thesis: verum