let x, y be object ; :: 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 over F
for P being RedSequence of ==>.-relation TS st P . 1 = [x,v] & P . (len P) = [y,w] holds
ex u being Element of E ^omega st v = u ^ 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 over F
for P being RedSequence of ==>.-relation TS st P . 1 = [x,v] & P . (len P) = [y,w] holds
ex u being Element of E ^omega st v = u ^ w
let v, w be Element of E ^omega ; :: thesis: for F being Subset of (E ^omega)
for TS being non empty transition-system over F
for P being RedSequence of ==>.-relation TS st P . 1 = [x,v] & P . (len P) = [y,w] holds
ex u being Element of E ^omega st v = u ^ w
let F be Subset of (E ^omega); :: thesis: for TS being non empty transition-system over F
for P being RedSequence of ==>.-relation TS st P . 1 = [x,v] & P . (len P) = [y,w] holds
ex u being Element of E ^omega st v = u ^ w
let TS be non empty transition-system over F; :: thesis: for P being RedSequence of ==>.-relation TS st P . 1 = [x,v] & P . (len P) = [y,w] holds
ex u being Element of E ^omega st v = u ^ w
let P be RedSequence of ==>.-relation TS; :: thesis: ( P . 1 = [x,v] & P . (len P) = [y,w] implies ex u being Element of E ^omega st v = u ^ w )
assume that
A1: P . 1 = [x,v] and
A2: P . (len P) = [y,w] ; :: thesis: ex u being Element of E ^omega st v = u ^ w
0 + 1 <= len P by NAT_1:8;
then 1 in dom P by FINSEQ_3:25;
then consider u being Element of E ^omega such that
A3: (P . 1) `2 = u ^ w by A2, Th52;
take u ; :: thesis: v = u ^ w
thus v = u ^ w by A1, A3; :: thesis: verum