let x1, x2, y1, y2 be set ; :: thesis: for E being non empty set
for F being Subset of (E ^omega )
for TS being transition-system of F st x1,x2 ==>. y1,y2,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 transition-system of F st x1,x2 ==>. y1,y2,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 transition-system of F st x1,x2 ==>. y1,y2,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 transition-system of F; :: thesis: ( x1,x2 ==>. y1,y2,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,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 ex v, w being Element of E ^omega st
( v = y2 & x1,w -->. y1,TS & x2 = w ^ v ) by DefDir;
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 ThProd30; :: thesis: verum