let x, y be object ; 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 over F st not <%> E in rng (dom the Tran of TS) holds
for P being RedSequence of ==>.-relation TS st P . 1 = [x,v] & P . (len P) = [y,w] & not len v > len w holds
( len P = 1 & x = y & v = w )
let E be 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 over F st not <%> E in rng (dom the Tran of TS) holds
for P being RedSequence of ==>.-relation TS st P . 1 = [x,v] & P . (len P) = [y,w] & not len v > len w holds
( len P = 1 & x = y & v = w )
let v, w be Element of E ^omega ; 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 st P . 1 = [x,v] & P . (len P) = [y,w] & not len v > len w holds
( len P = 1 & x = y & v = w )
let F be 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 st P . 1 = [x,v] & P . (len P) = [y,w] & not len v > len w holds
( len P = 1 & x = y & v = w )
let TS be non empty transition-system over F; ( not <%> E in rng (dom the Tran of TS) implies for P being RedSequence of ==>.-relation TS st P . 1 = [x,v] & P . (len P) = [y,w] & not len v > len w holds
( len P = 1 & x = y & v = w ) )
assume A1:
not <%> E in rng (dom the Tran of TS)
; for P being RedSequence of ==>.-relation TS st P . 1 = [x,v] & P . (len P) = [y,w] & not len v > len w holds
( len P = 1 & x = y & v = w )
let P be RedSequence of ==>.-relation TS; ( P . 1 = [x,v] & P . (len P) = [y,w] & not len v > len w implies ( len P = 1 & x = y & v = w ) )
assume A2:
( P . 1 = [x,v] & P . (len P) = [y,w] )
; ( len v > len w or ( len P = 1 & x = y & v = w ) )
consider u being Element of E ^omega such that
A3:
v = u ^ w
by A2, Th53;
A4:
len v >= len w
by A2, Th59;