let x1, x2, y1, y2 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 [[x1,x2],[y1,y2]] in ==>.-relation TS holds
( x1 in TS & y1 in TS & x2 in E ^omega & y2 in E ^omega & x1 in dom (dom the Tran of TS) & y1 in rng the Tran of TS )
let E be non empty set ; :: thesis: for F being Subset of (E ^omega )
for TS being non empty transition-system of F st [[x1,x2],[y1,y2]] in ==>.-relation TS holds
( x1 in TS & y1 in TS & x2 in E ^omega & y2 in E ^omega & x1 in dom (dom the Tran of TS) & y1 in rng the Tran of TS )
let F be Subset of (E ^omega ); :: thesis: for TS being non empty transition-system of F st [[x1,x2],[y1,y2]] in ==>.-relation TS holds
( x1 in TS & y1 in TS & x2 in E ^omega & y2 in E ^omega & x1 in dom (dom the Tran of TS) & y1 in rng the Tran of TS )
let TS be non empty transition-system of F; :: thesis: ( [[x1,x2],[y1,y2]] in ==>.-relation TS implies ( x1 in TS & y1 in TS & x2 in E ^omega & y2 in E ^omega & x1 in dom (dom the Tran of TS) & y1 in rng the Tran of TS ) )
assume [[x1,x2],[y1,y2]] in ==>.-relation TS ; :: thesis: ( x1 in TS & y1 in TS & x2 in E ^omega & y2 in E ^omega & x1 in dom (dom the Tran of TS) & y1 in rng the Tran of TS )
then x1,x2 ==>. y1,y2,TS by DefRel;
hence ( x1 in TS & y1 in TS & x2 in E ^omega & y2 in E ^omega & x1 in dom (dom the Tran of TS) & y1 in rng the Tran of TS ) by ThDir10; :: thesis: verum