let E be set ; :: thesis: for S, T being semi-Thue-system of E
for s, t being Element of E ^omega st S c= T & s ==>. t,S holds
s ==>. t,T
let S, T be semi-Thue-system of E; :: thesis: for s, t being Element of E ^omega st S c= T & s ==>. t,S holds
s ==>. t,T
let s, t be Element of E ^omega ; :: thesis: ( S c= T & s ==>. t,S implies s ==>. t,T )
assume that
A1: S c= T and
A2: s ==>. t,S ; :: thesis: s ==>. t,T
consider v, w, s1, t1 being Element of E ^omega such that
A3: ( s = (v ^ s1) ^ w & t = (v ^ t1) ^ w ) and
A4: s1 -->. t1,S by A2;
s1 -->. t1,T by A1, A4;
hence s ==>. t,T by A3; :: thesis: verum