let x1, x2, y1, y2 be object ; :: thesis: for E being non empty set
for F being Subset of (E ^omega)
for TS being non empty transition-system over F st [[x1,x2],[y1,y2]] in ==>.-relation TS holds
ex v, w being Element of E ^omega st
( v = y2 & x1,w -->. y1,TS & x2 = w ^ v )
let E be non empty set ; :: thesis: for F being Subset of (E ^omega)
for TS being non empty transition-system over F st [[x1,x2],[y1,y2]] in ==>.-relation TS holds
ex v, w being Element of E ^omega st
( v = y2 & x1,w -->. y1,TS & x2 = w ^ v )
let F be Subset of (E ^omega); :: thesis: for TS being non empty transition-system over F st [[x1,x2],[y1,y2]] in ==>.-relation TS holds
ex v, w being Element of E ^omega st
( v = y2 & x1,w -->. y1,TS & x2 = w ^ v )
let TS be non empty transition-system over F; :: thesis: ( [[x1,x2],[y1,y2]] in ==>.-relation TS implies ex v, w being Element of E ^omega st
( v = y2 & x1,w -->. y1,TS & x2 = w ^ v ) )
assume [[x1,x2],[y1,y2]] in ==>.-relation TS ; :: thesis: ex v, w being Element of E ^omega st
( v = y2 & x1,w -->. y1,TS & x2 = w ^ v )
then x1,x2 ==>. y1,y2,TS by Def4;
hence ex v, w being Element of E ^omega st
( v = y2 & x1,w -->. y1,TS & x2 = w ^ v ) ; :: thesis: verum