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 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 ; :: 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 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 ; :: thesis: for F being Subset of (E ^omega )
for TS being non empty transition-system of 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 ); :: thesis: for TS being non empty transition-system of 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 of F; :: thesis: ( 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 A: not <%> E in rng (dom the Tran of TS) ; :: thesis: 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; :: thesis: ( P . 1 = [x,v] & P . (len P) = [y,w] & not len v > len w implies ( len P = 1 & x = y & v = w ) )
assume B: ( P . 1 = [x,v] & P . (len P) = [y,w] ) ; :: thesis: ( len v > len w or ( len P = 1 & x = y & v = w ) )
consider u being Element of E ^omega such that
C: v = u ^ w by B, ThRedSeq60;
D: len v >= len w by B, ThRedSeq79;

per cases ( len v > len w or len v <= len w ) ;

suppose len v > len w ; :: thesis: ( len v > len w or ( len P = 1 & x = y & v = w ) )

hence ( len v > len w or ( len P = 1 & x = y & v = w ) ) ; :: thesis: verum

end;

suppose len v <= len w ; :: thesis: ( len v > len w or ( len P = 1 & x = y & v = w ) )

then D1: len v = len w by D, XXREAL_0:1;
len v = (len u) + (len w) by C, AFINSQ_1:20;
then u = <%> E by D1;
then v = w by C, AFINSQ_1:32;
hence ( len v > len w or ( len P = 1 & x = y & v = w ) ) by A, B, ThRedSeq80; :: thesis: verum

end;

end;