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 Th22:
  S c= ==>.-relation(S)
proof
 let x be object;
    assume
A1: x in S;
    then consider a, b being object such that
A2: a in E^omega & b in E^omega and
A3: x = [a, b] by ZFMISC_1:def 2;
    reconsider a, b as Element of E^omega by A2;
    a -->. b, S by A1,A3;
    then a ==>.b, S by Th10;
    hence thesis by A3,Def6;
end;
