reserve x,y for object,X for set,
  f for Function,
  R,S for Relation;

theorem Th9:
  f is real-valued iff for x holds f.x is real
proof
  hereby
    assume
A1: f is real-valued;
    let x;
    per cases;
    suppose
      x in dom f;
      hence f.x is real by A1;
    end;
    suppose
      not x in dom f;
      hence f.x is real by FUNCT_1:def 2;
    end;
  end;
  thus thesis;
end;
