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 Th123:
  doms(k,n)^doms(k,m) = doms(k,n+m)
proof
  thus doms(k,n)^doms(k,m) c= doms(k,n+m)
  proof
    let y;
    assume y in doms(k,n)^doms(k,m);
    then ex p,s be FinSequence st y=p^s & p in doms(k,n) & s in doms(k,m)
      by POLNOT_1:def 2;
    hence thesis by Th122;
  end;
  let y;
  assume
A1: y in doms(k,n+m);
  then consider p be Element of (Seg k)* such that
A2: p=y & len p= n+m;
  consider q be FinSequence such that
A3: p=(p|n)^q by FINSEQ_1:80;
  len (p|n) =n by NAT_1:11,A2,FINSEQ_1:59;
  then p|n in doms(k,n) & q in doms(k,m) by A1,A2,A3,Th120;
  hence thesis by A2,A3,POLNOT_1:def 2;
end;
