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

proof

assume x,y -->. z,TS ; :: thesis: [[x,y],[z,(<%> E)]] in ==>.-relation TS
then x,y ==>. z, <%> E,TS by Th23;
hence [[x,y],[z,(<%> E)]] in ==>.-relation TS by Def4; :: thesis: verum

end;

assume [[x,y],[z,(<%> E)]] in ==>.-relation TS ; :: thesis: x,y -->. z,TS
then x,y ==>. z, <%> E,TS by Def4;
hence x,y -->. z,TS by Th23; :: thesis: verum