theorem Th39:
  for f, g being Function of X,TOP-REAL n holds
  f<++>g is Function of X,TOP-REAL n
  proof
    let f, g be Function of X,TOP-REAL n;
    set h = f<++>g;
A1: dom f = X by FUNCT_2:def 1;
 dom g = X by FUNCT_2:def 1;
then A2: dom h = X by A1,VALUED_2:def 45;
    for x st x in X holds h.x in the carrier of TOP-REAL n
    proof
      let x;
      assume
A3:   x in X;
      then reconsider X as non empty set;
      reconsider x as Element of X by A3;
      reconsider f, g as Function of X,TOP-REAL n;
A4:   ((f.x) qua real-valued Function)+g.x = f.x+g.x;
      h.x = ((f.x) qua real-valued Function)+g.x by A2,VALUED_2:def 45;
      hence thesis by A4;
    end;
    hence thesis by A2,FUNCT_2:3;
  end;
