theorem Th21:
  p = (p ^ q) | (dom p)
proof
A1: dom(p ^ q) = Seg(len p + len q) by Def7;
A2: dom p = Seg len p by Def3; then
A3: dom p = dom(p ^ q) /\ Seg len p by A1,Th7,NAT_1:12;
  for x being object st x in dom p holds p.x = (p ^ q).x by Def7;
  hence thesis by A2,A3,FUNCT_1:46;
end;
