let E be set ; :: thesis: for S being semi-Thue-system of E
for s, t being Element of E ^omega st s ==>. t,S holds
s ==>* t,S
let S be semi-Thue-system of E; :: thesis: for s, t being Element of E ^omega st s ==>. t,S holds
s ==>* t,S
let s, t be Element of E ^omega ; :: thesis: ( s ==>. t,S implies s ==>* t,S )
assume s ==>. t,S ; :: thesis: s ==>* t,S
then [s,t] in ==>.-relation S by Def6;
then ==>.-relation S reduces s,t by REWRITE1:15;
hence s ==>* t,S by Def7; :: thesis: verum