
theorem
  for V being RealLinearSpace, M being Subset of V, r being Real holds r
  *Cir M = Cir(r*M)
proof
  let V be RealLinearSpace, M be Subset of V, r be Real;
  thus r*Cir M c= Cir(r*M)
  proof
    let x be object;
    per cases;
    suppose
A1:   r = 0;
      per cases;
      suppose
        M = {};
        then M = {}V;
        then Cir M = {} by Th14;
        hence thesis by CONVEX1:33;
      end;
      suppose
A2:     M <> {};
        then r * M = {0.V} by A1,CONVEX1:34;
        then
A3:     {0.V} c= Cir(r*M) by Th11;
        Cir M <> {} by A2,Th11,XBOOLE_1:3;
        then r*Cir M = {0.V} by A1,CONVEX1:34;
        hence thesis by A3;
      end;
    end;
    suppose
A4:   r <> 0;
A5:   r*Cir M = {r*v where v is Point of V: v in Cir M} by CONVEX1:def 1;
      assume x in r*Cir M;
      then consider v being Point of V such that
A6:   x = r*v and
A7:   v in Cir M by A5;
      for W being set st W in Circled-Family (r*M) holds r*v in W
      proof
        let W be set;
        assume
A8:     W in Circled-Family (r*M);
        then reconsider W as Subset of V;
        r * M c= W by A8,Def2;
        then r"*(r*M) c= r"*W by CONVEX1:39;
        then (r"*r)*M c= r"*W by CONVEX1:37;
        then 1*M c= r"*W by A4,XCMPLX_0:def 7;
        then
A9:     M c= r"*W by CONVEX1:32;
        W is circled by A8,Def2;
        then r"*W is circled by Th2;
        then r"*W in Circled-Family M by A9,Def2;
        then r"*W = {r"*w where w is Point of V: w in W} & v in r"*W by A7,
CONVEX1:def 1,SETFAM_1:def 1;
        then consider w being Point of V such that
A10:    v = r"*w and
A11:    w in W;
        r*v = (r*r")*w by A10,RLVECT_1:def 7
          .= 1*w by A4,XCMPLX_0:def 7
          .= w by RLVECT_1:def 8;
        hence thesis by A11;
      end;
      hence thesis by A6,SETFAM_1:def 1;
    end;
  end;
  r * M c= r*Cir M & r*Cir M is circled by Th2,Th11,CONVEX1:39;
  hence thesis by Th12;
end;
