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 transition-system of F st x,u ==>. y,v,TS holds
x,u ^ w ==>. y,v ^ w,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 transition-system of F st x,u ==>. y,v,TS holds
x,u ^ w ==>. y,v ^ w,TS
let u, v, w be Element of E ^omega ; :: thesis: for F being Subset of (E ^omega )
for TS being transition-system of F st x,u ==>. y,v,TS holds
x,u ^ w ==>. y,v ^ w,TS
let F be Subset of (E ^omega ); :: thesis: for TS being transition-system of F st x,u ==>. y,v,TS holds
x,u ^ w ==>. y,v ^ w,TS
let TS be transition-system of F; :: thesis: ( x,u ==>. y,v,TS implies x,u ^ w ==>. y,v ^ w,TS )
assume x,u ==>. y,v,TS ; :: thesis: x,u ^ w ==>. y,v ^ w,TS
then consider u1 being Element of E ^omega such that
A: ( x,u1 -->. y,TS & u = u1 ^ v ) by ThDir25;
u ^ w = u1 ^ (v ^ w) by A, AFINSQ_1:30;
hence x,u ^ w ==>. y,v ^ w,TS by A, DefDir; :: thesis: verum