let x, y be set ; :: thesis: for E being non empty set
for v, w being Element of E ^omega
for F being Subset of (E ^omega )
for TS being non empty transition-system of F holds
( <%> E in rng (dom the Tran of TS) or not ==>.-relation TS reduces [x,v],[y,w] or len v > len w or ( x = y & v = w ) )
let E be non empty set ; :: thesis: for v, w being Element of E ^omega
for F being Subset of (E ^omega )
for TS being non empty transition-system of F holds
( <%> E in rng (dom the Tran of TS) or not ==>.-relation TS reduces [x,v],[y,w] or len v > len w or ( x = y & v = w ) )
let v, w be Element of E ^omega ; :: thesis: for F being Subset of (E ^omega )
for TS being non empty transition-system of F holds
( <%> E in rng (dom the Tran of TS) or not ==>.-relation TS reduces [x,v],[y,w] or len v > len w or ( x = y & v = w ) )
let F be Subset of (E ^omega ); :: thesis: for TS being non empty transition-system of F holds
( <%> E in rng (dom the Tran of TS) or not ==>.-relation TS reduces [x,v],[y,w] or len v > len w or ( x = y & v = w ) )
let TS be non empty transition-system of F; :: thesis: ( <%> E in rng (dom the Tran of TS) or not ==>.-relation TS reduces [x,v],[y,w] or len v > len w or ( x = y & v = w ) )
assume A: not <%> E in rng (dom the Tran of TS) ; :: thesis: ( not ==>.-relation TS reduces [x,v],[y,w] or len v > len w or ( x = y & v = w ) )
assume ==>.-relation TS reduces [x,v],[y,w] ; :: thesis: ( len v > len w or ( x = y & v = w ) )
then consider P being RedSequence of ==>.-relation TS such that
B: ( P . 1 = [x,v] & P . (len P) = [y,w] ) by REWRITE1:def 3;
thus ( len v > len w or ( x = y & v = w ) ) by A, B, ThRedSeq110; :: thesis: verum