reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;

theorem Th61:
  for p being FuncSequence holds dom compose(p,firstdom p) =
  firstdom p
proof
  let p be FuncSequence;
  per cases;
  suppose
    p = {};
    then compose(p,firstdom p) = id firstdom p by Th38;
    hence thesis;
  end;
  suppose
    p <> {};
    then dom compose(p,firstdom p) = (firstdom p) /\ firstdom p by Th60;
    hence thesis;
  end;
end;
