reserve n for Nat,
  a, b, r, w for Real,
  x, y, z for Point of TOP-REAL n,
  e for Point of Euclid n;
reserve V for RealLinearSpace,
        p,q,x for Element of V;
reserve p, q, x for Point of TOP-REAL n;
reserve s, t for Point of TOP-REAL 2;

theorem Th63:
 for V being RealLinearSpace, p1,p2 being Point of V
  holds Line(p1,p2) = halfline(p1,p2) \/ halfline(p2,p1)
proof
 let V be RealLinearSpace, p1,p2 be Point of V;
 thus Line(p1,p2) c= halfline(p1,p2) \/ halfline(p2,p1)
  proof let e be object;
   assume e in Line(p1,p2);
    then consider r such that
A1:   e = (1-r)*p1 + r*p2;
    now per cases;
     case r >= 0;
      hence e in halfline(p1,p2) by A1;
     end;
     case r <= 0;
      then
A2:     1-r >= 0;
       e = (1-(1-r))*p2 + (1-r)*p1 by A1;
      hence e in halfline(p2,p1) by A2;
     end;
    end;
   hence thesis by XBOOLE_0:def 3;
  end;
  halfline(p1,p2) c= Line(p1,p2) & halfline(p2,p1) c= Line(p1,p2) by Th62;
 hence thesis by XBOOLE_1:8;
end;
