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 Th120:
  for f,g be FinSequence st
    (len f=n or len g = m) & f^g in doms(k,n+m) holds
  f in doms(k,n) & g in doms(k,m)
proof
  let f,g be FinSequence;
  set fg=f^g;
  assume
A1: (len f=n or len g = m) & f^g in doms(k,n+m);
  then consider s be Element of (Seg k)* such that
A2: f^g = s & len s = n+m;
  reconsider s as FinSequence of Seg k;
A3: len (f^g) = len f+len g by FINSEQ_1:22;
  rng f c= rng s & rng g c= rng s & rng s c= Seg k
     by A2,FINSEQ_1:30,29;
  then rng f c= Seg k & rng g c= Seg k;
  then f is FinSequence of Seg k & g is FinSequence of Seg k
    by FINSEQ_1:def 4;
  then f in (Seg k)* & g in (Seg k)* by FINSEQ_1:def 11;
  hence thesis by A3,A1,A2;
end;
