let x, y be object ; 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 over 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 ; for u, v, w being Element of E ^omega
for F being Subset of (E ^omega)
for TS being non empty transition-system over 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 ; for F being Subset of (E ^omega)
for TS being non empty transition-system over 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); for TS being non empty transition-system over 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 over F; ( [[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
; [[x,(u ^ w)],[y,(v ^ w)]] in ==>.-relation TS
then
x,u ==>. y,v,TS
by Def4;
then
x,u ^ w ==>. y,v ^ w,TS
by Th25;
hence
[[x,(u ^ w)],[y,(v ^ w)]] in ==>.-relation TS
by Def4; verum