let x, y be set ; :: thesis: for E being non empty set
for u, v, w being Element of E ^omega
for F being Subset of (E ^omega )
for TS being non empty transition-system of F st [[x,u],[y,v]] in ==>.-relation TS holds
[[x,(u ^ w)],[y,(v ^ w)]] in ==>.-relation TS
let E be non empty set ; :: thesis: for u, v, w being Element of E ^omega
for F being Subset of (E ^omega )
for TS being non empty transition-system of F st [[x,u],[y,v]] in ==>.-relation TS holds
[[x,(u ^ w)],[y,(v ^ w)]] in ==>.-relation TS
let u, v, w be Element of E ^omega ; :: thesis: for F being Subset of (E ^omega )
for TS being non empty transition-system of F st [[x,u],[y,v]] in ==>.-relation TS holds
[[x,(u ^ w)],[y,(v ^ w)]] in ==>.-relation TS
let F be Subset of (E ^omega ); :: thesis: for TS being non empty transition-system of F st [[x,u],[y,v]] in ==>.-relation TS holds
[[x,(u ^ w)],[y,(v ^ w)]] in ==>.-relation TS
let TS be non empty transition-system of F; :: thesis: ( [[x,u],[y,v]] in ==>.-relation TS implies [[x,(u ^ w)],[y,(v ^ w)]] in ==>.-relation TS )
assume [[x,u],[y,v]] in ==>.-relation TS ; :: thesis: [[x,(u ^ w)],[y,(v ^ w)]] in ==>.-relation TS
then x,u ==>. y,v,TS by DefRel;
then x,u ^ w ==>. y,v ^ w,TS by ThDir30;
hence [[x,(u ^ w)],[y,(v ^ w)]] in ==>.-relation TS by DefRel; :: thesis: verum