let E be set ; :: thesis: for S being semi-Thue-system of E
for w being Element of E ^omega holds Lang (w,S) = Lang (w,(S \/ (id (E ^omega))))
let S be semi-Thue-system of E; :: thesis: for w being Element of E ^omega holds Lang (w,S) = Lang (w,(S \/ (id (E ^omega))))
let w be Element of E ^omega ; :: thesis: Lang (w,S) = Lang (w,(S \/ (id (E ^omega))))
A1: Lang (w,(S \/ (id (E ^omega)))) c= Lang (w,S)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Lang (w,(S \/ (id (E ^omega)))) or x in Lang (w,S) )
assume A2: x in Lang (w,(S \/ (id (E ^omega)))) ; :: thesis: x in Lang (w,S)
then reconsider s = x as Element of E ^omega ;
w ==>* s,S \/ (id (E ^omega)) by A2, Th46;
then w ==>* s,S by Th41;
hence x in Lang (w,S) ; :: thesis: verum

end;

Lang (w,S) c= Lang (w,(S \/ (id (E ^omega)))) by Th48, XBOOLE_1:7;
hence Lang (w,S) = Lang (w,(S \/ (id (E ^omega)))) by A1, XBOOLE_0:def 10; :: thesis: verum