theorem Th15:
  f tolerates g & x in dom f implies (f+*g).x = f.x
proof
  assume that
A1: f tolerates g and
A2: x in dom f;
  now
    per cases;
    suppose
      x in dom g;
      then x in dom f /\ dom g & (f+*g).x = g.x by A2,Th13,XBOOLE_0:def 4;
      hence thesis by A1;
    end;
    suppose
      not x in dom g;
      hence thesis by Th11;
    end;
  end;
  hence thesis;
end;
