reserve r,s,t,u for Real;

theorem Th18:
  for X being RealLinearSpace, M being Subset of X, r being Real
  holds r*conv(M) = conv(r*M)
proof
  let X be RealLinearSpace, M be Subset of X, r be Real;
  thus r*conv(M) c= conv(r*M)
  proof
    let x be object;
    per cases;
    suppose
A1:   r = 0;
      per cases;
      suppose
        M = {};
        hence thesis by CONVEX1:33;
      end;
      suppose
A2:     M <> {};
        then r*M = {0.X} by A1,CONVEX1:34;
        then
A3:     {0.X} c= conv(r*M) by CONVEX1:41;
        conv(M) <> {} by A2,CONVEX1:41,XBOOLE_1:3;
        then r*conv(M) = {0.X} by A1,CONVEX1:34;
        hence thesis by A3;
      end;
    end;
    suppose
A4:   r <> 0;
      assume x in r*conv(M);
      then consider v being Point of X such that
A5:   x = r*v and
A6:   v in conv(M);
      for V being set st V in Convex-Family(r*M) holds r*v in V
      proof
        let V be set;
        assume
A7:     V in Convex-Family(r*M);
        then reconsider V as Subset of X;
        r*M c= V by A7,CONVEX1:def 4;
        then r"*(r*M) c= r"*V by CONVEX1:39;
        then (r"*r)*M c= r"*V by CONVEX1:37;
        then 1*M c= r"*V by A4,XCMPLX_0:def 7;
        then
A8:     M c= r"*V by CONVEX1:32;
        V is convex by A7,CONVEX1:def 4;
        then r"*V is convex by CONVEX1:1;
        then r"*V in Convex-Family M by A8,CONVEX1:def 4;
        then v in r"*V by A6,SETFAM_1:def 1;
        then consider w being Point of X such that
A9:     v = r"*w and
A10:    w in V;
        r*v = (r*r")*w by A9,RLVECT_1:def 7
          .= 1*w by A4,XCMPLX_0:def 7
          .= w by RLVECT_1:def 8;
        hence thesis by A10;
      end;
      hence thesis by A5,SETFAM_1:def 1;
    end;
  end;
  r*M c= r*conv(M) & r*conv(M) is convex by CONVEX1:1,39,41;
  hence thesis by CONVEX1:30;
end;
