let x, y be set ; :: thesis: for E being non empty set
for u, v 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
ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
let E be non empty set ; :: thesis: for u, v 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
ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
let u, v 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
ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
let F be Subset of (E ^omega); :: thesis: for TS being transition-system of F st x,u ==>. y,v,TS holds
ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
let TS be transition-system of F; :: thesis: ( x,u ==>. y,v,TS implies ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v ) )
assume x,u ==>. y,v,TS ; :: thesis: ex w being Element of E ^omega st
( x,w -->. y,TS & u = w ^ v )
then consider v1, w being Element of E ^omega such that
A1: ( v1 = v & x,w -->. y,TS & u = w ^ v1 ) by Def3;
take w ; :: thesis: ( x,w -->. y,TS & u = w ^ v )
thus ( x,w -->. y,TS & u = w ^ v ) by A1; :: thesis: verum