let E be set ; :: thesis: for S, T being semi-Thue-system of E
for s, t being Element of E ^omega holds
( not s ==>. t,S \/ T or s ==>. t,S or s ==>. t,T )
let S, T be semi-Thue-system of E; :: thesis: for s, t being Element of E ^omega holds
( not s ==>. t,S \/ T or s ==>. t,S or s ==>. t,T )
let s, t be Element of E ^omega ; :: thesis: ( not s ==>. t,S \/ T or s ==>. t,S or s ==>. t,T )
assume s ==>. t,S \/ T ; :: thesis: ( s ==>. t,S or s ==>. t,T )
then consider v, w, s1, t1 being Element of E ^omega such that
A1: ( s = (v ^ s1) ^ w & t = (v ^ t1) ^ w & s1 -->. t1,S \/ T ) by Def5;
A2: [s1,t1] in S \/ T by A1, Def4;

per cases ( [s1,t1] in S or [s1,t1] in T ) by A2, XBOOLE_0:def 3;

suppose [s1,t1] in S ; :: thesis: ( s ==>. t,S or s ==>. t,T )

then s1 -->. t1,S by Def4;
hence ( s ==>. t,S or s ==>. t,T ) by A1, Def5; :: thesis: verum

end;

suppose [s1,t1] in T ; :: thesis: ( s ==>. t,S or s ==>. t,T )

then s1 -->. t1,T by Def4;
hence ( s ==>. t,S or s ==>. t,T ) by A1, Def5; :: thesis: verum

end;

end;