reserve x for set;
reserve k, l for Nat;
reserve p, q for FinSequence;
reserve R for Relation;
reserve p, q for RedSequence of R;
reserve E for set;
reserve s, t for XFinSequence;
reserve p, q for XFinSequence-yielding FinSequence;
reserve E for set;
reserve S, T, U for semi-Thue-system of E;
reserve s, t, s1, t1, u, v, u1, v1, w for Element of E^omega;
reserve p for FinSequence of E^omega;

theorem Th26:
  S c= T implies ==>.-relation(S) c= ==>.-relation(T)
proof
  assume
A1: S c= T;
 let x be object;
    assume
A2: x in ==>.-relation(S);
    then consider s, t being object such that
A3: x = [s, t] and
A4: s in E^omega & t in E^omega by RELSET_1:2;
    reconsider s, t as Element of E^omega by A4;
    s ==>. t, S by A2,A3,Def6;
    then s ==>. t, T by A1,Th19;
    hence thesis by A3,Def6;
end;
