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;
