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 holds
( x,v -->. y,TS iff <*[x,(v ^ w)],[y,w]*> is RedSequence of ==>.-relation TS )
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 holds
( x,v -->. y,TS iff <*[x,(v ^ w)],[y,w]*> is RedSequence of ==>.-relation TS )
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 holds
( x,v -->. y,TS iff <*[x,(v ^ w)],[y,w]*> is RedSequence of ==>.-relation TS )
let F be Subset of (E ^omega); :: thesis: for TS being non empty transition-system over F holds
( x,v -->. y,TS iff <*[x,(v ^ w)],[y,w]*> is RedSequence of ==>.-relation TS )
let TS be non empty transition-system over F; :: thesis: ( x,v -->. y,TS iff <*[x,(v ^ w)],[y,w]*> is RedSequence of ==>.-relation TS )
thus ( x,v -->. y,TS implies <*[x,(v ^ w)],[y,w]*> is RedSequence of ==>.-relation TS ) :: thesis: ( <*[x,(v ^ w)],[y,w]*> is RedSequence of ==>.-relation TS implies x,v -->. y,TS )

proof

assume x,v -->. y,TS ; :: thesis: <*[x,(v ^ w)],[y,w]*> is RedSequence of ==>.-relation TS
then [[x,(v ^ w)],[y,w]] in ==>.-relation TS by Th38;
hence <*[x,(v ^ w)],[y,w]*> is RedSequence of ==>.-relation TS by REWRITE1:7; :: thesis: verum

end;

assume <*[x,(v ^ w)],[y,w]*> is RedSequence of ==>.-relation TS ; :: thesis: x,v -->. y,TS
then [[x,(v ^ w)],[y,w]] in ==>.-relation TS by Th8;
hence x,v -->. y,TS by Th38; :: thesis: verum