reserve x,r,a for Real;
reserve f,g for Function,
  x1,x2 for set;

theorem
  f.:(dom f /\ dom g) c= rng g implies rng f \/ rng g = rng(f+*g)
proof
  assume f.:(dom f /\ dom g) c= rng g;
  then
A1: rng f \ rng g c= rng(f+*g) by Lm1;
  rng g c= rng(f +* g) by FUNCT_4:18;
  then (rng f \ rng g) \/ rng g c= rng(f +* g) by A1,XBOOLE_1:8;
  then
A2: rng f \/ rng g c= rng(f +* g) by XBOOLE_1:39;
  rng(f+*g) c= rng f \/ rng g by FUNCT_4:17;
  hence thesis by A2,XBOOLE_0:def 10;
end;
