let E be set ; :: thesis: for S, T being semi-Thue-system of E
for w being Element of E ^omega st S,T are_equivalent_wrt w holds
S,S \/ T are_equivalent_wrt w
let S, T be semi-Thue-system of E; :: thesis: for w being Element of E ^omega st S,T are_equivalent_wrt w holds
S,S \/ T are_equivalent_wrt w
let w be Element of E ^omega ; :: thesis: ( S,T are_equivalent_wrt w implies S,S \/ T are_equivalent_wrt w )
assume A1: S,T are_equivalent_wrt w ; :: thesis: S,S \/ T are_equivalent_wrt w
thus Lang w,S = Lang w,(S \/ T) :: according to REWRITE2:def 9 :: thesis: verum

proof

A2: Lang w,S c= Lang w,(S \/ T) by Th48, XBOOLE_1:7;
Lang w,(S \/ T) c= Lang w,S

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in Lang w,(S \/ T) or x in Lang w,S )
assume A3: x in Lang w,(S \/ T) ; :: thesis: x in Lang w,S
reconsider s = x as Element of E ^omega by A3;
w ==>* s,S \/ T by A3, Th46;
then w ==>* s,S by A1, Th59;
hence x in Lang w,S ; :: thesis: verum

end;

hence Lang w,S = Lang w,(S \/ T) by A2, XBOOLE_0:def 10; :: thesis: verum

end;