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

proof

assume x,y -->. z,TS ; :: thesis: <*[x,y],[z,(<%> E)]*> is RedSequence of ==>.-relation TS
then [[x,y],[z,(<%> E)]] in ==>.-relation TS by Th37;
hence <*[x,y],[z,(<%> E)]*> is RedSequence of ==>.-relation TS by REWRITE1:7; :: thesis: verum

end;

assume <*[x,y],[z,(<%> E)]*> is RedSequence of ==>.-relation TS ; :: thesis: x,y -->. z,TS
then [[x,y],[z,(<%> E)]] in ==>.-relation TS by Th8;
hence x,y -->. z,TS by Th37; :: thesis: verum