reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th1:
  for f,g being Function st f tolerates g for A being set holds (f
  +*g)"A = (f"A)\/(g"A)
proof
  let f,g be Function;
  assume
A1: f tolerates g;
  let A be set;
  f c= f+*g by A1,FUNCT_4:28;
  then
A2: f"A c= (f+*g)"A by RELAT_1:144;
  thus (f+*g)"A c= (f"A)\/(g"A)
  proof
    let x be object;
    assume
A3: x in (f+*g)"A;
    then x in dom (f+*g) by FUNCT_1:def 7;
    then x in dom f \/ dom g by FUNCT_4:def 1;
    then x in dom f or x in dom g by XBOOLE_0:def 3;
    then
A4: x in dom f & (f+*g).x = f.x or x in dom g & (f+*g).x = g.x by A1,FUNCT_4:13
,15;
    (f+*g).x in A by A3,FUNCT_1:def 7;
    then x in f"A or x in g"A by A4,FUNCT_1:def 7;
    hence thesis by XBOOLE_0:def 3;
  end;
  g"A c= (f+*g)"A by FUNCT_4:25,RELAT_1:144;
  hence thesis by A2,XBOOLE_1:8;
end;
