reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;

theorem Th13:
  for f being quasi_total Relation of X,Y for g being quasi_total
  Relation of Y,Z st Y = {} implies Z = {} or X = {} holds f*g is quasi_total
proof
  let f be quasi_total Relation of X,Y;
  let g be quasi_total Relation of Y,Z such that
A1: Y = {} implies Z = {} or X = {};
  per cases;
  case
A2: Z <> {};
    then
A3: dom g = Y by Def1;
    dom f = X & rng f c= Y by A1,A2,Def1;
    hence thesis by A3,RELAT_1:27;
  end;
  case Z = {};
    hence thesis;
  end;
end;
