
theorem Th2:
  for V be RealLinearSpace, M being Subset of V, r being Real st M
  is circled holds r * M is circled
proof
  let V be RealLinearSpace, M be Subset of V, r be Real;
  assume
A1: M is circled;
  for u being VECTOR of V, p being Real st |.p.| <= 1 & u in r * M holds p
  *u in r*M
  proof
    let u be VECTOR of V, p be Real;
    assume that
A2: |.p.| <= 1 and
A3: u in r * M;
    u in {r * w where w is Element of V: w in M} by A3,CONVEX1:def 1;
    then consider u9 be Element of V such that
A4: u = r * u9 and
A5: u9 in M;
A6: p*u = r*p*u9 by A4,RLVECT_1:def 7
      .= r*(p*u9) by RLVECT_1:def 7;
    p*u9 in M by A1,A2,A5;
    hence thesis by A6,RLTOPSP1:1;
  end;
  hence thesis;
end;
