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

proof

assume A1: x,y -->. z,TS ; :: thesis: x,y ==>. z, <%> E,TS
then y in F by Th15;
then reconsider w = y as Element of E ^omega ;
w = w ^ {} ;
hence x,y ==>. z, <%> E,TS by A1; :: thesis: verum

end;

assume x,y ==>. z, <%> E,TS ; :: thesis: x,y -->. z,TS
then ex v, w being Element of E ^omega st
( v = <%> E & x,w -->. z,TS & y = w ^ v ) ;
hence x,y -->. z,TS ; :: thesis: verum