let x, y 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 st not <%> E in rng (dom the Tran of TS) holds
( x, <%> E ==>* y,TS iff x = y )
let E be non empty set ; :: thesis: for F being Subset of (E ^omega )
for TS being non empty transition-system of F st not <%> E in rng (dom the Tran of TS) holds
( x, <%> E ==>* y,TS iff x = y )
let F be Subset of (E ^omega ); :: thesis: for TS being non empty transition-system of F st not <%> E in rng (dom the Tran of TS) holds
( x, <%> E ==>* y,TS iff x = y )
let TS be non empty transition-system of F; :: thesis: ( not <%> E in rng (dom the Tran of TS) implies ( x, <%> E ==>* y,TS iff x = y ) )
assume A1: not <%> E in rng (dom the Tran of TS) ; :: thesis: ( x, <%> E ==>* y,TS iff x = y )
thus ( x, <%> E ==>* y,TS implies x = y ) :: thesis: ( x = y implies x, <%> E ==>* y,TS )

proof

assume x, <%> E ==>* y,TS ; :: thesis: x = y
then x, <%> E ==>* y, <%> E,TS by Def7;
hence x = y by A1, Th91; :: thesis: verum

end;

assume x = y ; :: thesis: x, <%> E ==>* y,TS
hence x, <%> E ==>* y,TS by Th95; :: thesis: verum