let x, y be set ; :: thesis: for E being non empty set
for u being Element of E ^omega
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,u ==>* y,u,TS iff x = y )
let E be non empty set ; :: thesis: for u being Element of E ^omega
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,u ==>* y,u,TS iff x = y )
let u be Element of E ^omega ; :: 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,u ==>* y,u,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,u ==>* y,u,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,u ==>* y,u,TS iff x = y ) )
assume A: not <%> E in rng (dom the Tran of TS) ; :: thesis: ( x,u ==>* y,u,TS iff x = y )
thus ( x,u ==>* y,u,TS implies x = y ) :: thesis: ( x = y implies x,u ==>* y,u,TS )

proof

assume x,u ==>* y,u,TS ; :: thesis: x = y
then ==>.-relation TS reduces [x,u],[y,u] by DefTran;
then consider p being RedSequence of ==>.-relation TS such that
B: ( p . 1 = [x,u] & p . (len p) = [y,u] ) by REWRITE1:def 3;
thus x = y by A, B, ThRedSeq80; :: thesis: verum

end;

assume x = y ; :: thesis: x,u ==>* y,u,TS
hence x,u ==>* y,u,TS by ThTran10; :: thesis: verum