let x, y be object ; :: thesis: for E being non empty set
for v, w being Element of E ^omega
for F being Subset of (E ^omega)
for TS being non empty transition-system over F holds
( x,v -->. y,TS iff [[x,(v ^ w)],[y,w]] in ==>.-relation TS )
let E be non empty set ; :: thesis: for v, w being Element of E ^omega
for F being Subset of (E ^omega)
for TS being non empty transition-system over F holds
( x,v -->. y,TS iff [[x,(v ^ w)],[y,w]] in ==>.-relation TS )
let v, w be Element of E ^omega ; :: thesis: for F being Subset of (E ^omega)
for TS being non empty transition-system over F holds
( x,v -->. y,TS iff [[x,(v ^ w)],[y,w]] in ==>.-relation TS )
let F be Subset of (E ^omega); :: thesis: for TS being non empty transition-system over F holds
( x,v -->. y,TS iff [[x,(v ^ w)],[y,w]] in ==>.-relation TS )
let TS be non empty transition-system over F; :: thesis: ( x,v -->. y,TS iff [[x,(v ^ w)],[y,w]] in ==>.-relation TS )
thus ( x,v -->. y,TS implies [[x,(v ^ w)],[y,w]] in ==>.-relation TS ) :: thesis: ( [[x,(v ^ w)],[y,w]] in ==>.-relation TS implies x,v -->. y,TS )

proof

assume x,v -->. y,TS ; :: thesis: [[x,(v ^ w)],[y,w]] in ==>.-relation TS
then x,v ^ w ==>. y,w,TS ;
hence [[x,(v ^ w)],[y,w]] in ==>.-relation TS by Def4; :: thesis: verum

end;

assume [[x,(v ^ w)],[y,w]] in ==>.-relation TS ; :: thesis: x,v -->. y,TS
then x,v ^ w ==>. y,w,TS by Def4;
hence x,v -->. y,TS by Th24; :: thesis: verum