reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;

theorem Th3:
  rng S1 c= rng(S1^S2) & rng S2 c= rng(S1^S2)
  proof
    set q = S1^S2;
A1: dom q = (dom S1)+^dom S2 by ORDINAL4:def 1; then
A2: dom S1 c= dom q by ORDINAL3:24;
    thus rng S1 c= rng(S1^S2)
    proof
      let x be object; assume x in rng S1; then
      consider z being object such that
A3:   z in dom S1 & x = S1.z by FUNCT_1:def 3;
      reconsider z as Ordinal by A3;
      q.z = x & z in dom q by A3,A2,ORDINAL4:def 1;
      hence thesis by FUNCT_1:def 3;
    end;
    let x be object; assume x in rng S2; then
    consider z being object such that
A4: z in dom S2 & x = S2.z by FUNCT_1:def 3;
    reconsider z as Ordinal by A4;
    q.((dom S1)+^z) = x & (dom S1)+^z in dom q
    by A1,A4,ORDINAL3:17,ORDINAL4:def 1;
    hence thesis by FUNCT_1:def 3;
  end;
