reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;
reserve
  B,A,M for BinOp of D,
  F,G for D* -valued FinSequence,
  f for FinSequence of D,
  d,d1,d2 for Element of D;
reserve
  F,G for non-empty non empty FinSequence of D*,
  f for non empty FinSequence of D;
reserve f,g for FinSequence of D,
        a,b,c for set,
        F,F1,F2 for finite set;

theorem Th125:
  k <= n implies doms(k,m) c= doms(n,m)
proof
  assume k <= n;
  then
A1: Seg k c= Seg n by FINSEQ_1:5;
  let y;
  assume y in doms(k,m);
  then consider p be Element of (Seg k)* such that
A2: y=p & len p = m;
  rng p c= Seg n by A1;
  then p is FinSequence of Seg n by FINSEQ_1:def 4;
  then p in (Seg n)* by FINSEQ_1:def 11;
  hence thesis by A2;
end;
