
theorem Th3:
  for V be RealLinearSpace, M1,M2 being Subset of V, r1,r2 being
  Real st M1 is circled & M2 is circled holds r1*M1 + r2*M2 is circled
proof
  let V be RealLinearSpace, M1,M2 be Subset of V, r1,r2 be Real;
  assume that
A1: M1 is circled and
A2: M2 is circled;
  let u be VECTOR of V, p be Real;
  assume that
A3: |.p.| <= 1 and
A4: u in r1*M1 + r2*M2;
  u in {x+y where x,y is VECTOR of V : x in r1*M1 & y in r2*M2} by A4,
RUSUB_4:def 9;
  then consider u1,u2 be VECTOR of V such that
A5: u = u1 + u2 and
A6: u1 in r1*M1 and
A7: u2 in r2*M2;
  u1 in {r1*x where x is VECTOR of V : x in M1} by A6,CONVEX1:def 1;
  then consider x1 be VECTOR of V such that
A8: u1 = r1*x1 and
A9: x1 in M1;
A10: p*u1 = r1*p*x1 by A8,RLVECT_1:def 7
    .= r1*(p*x1) by RLVECT_1:def 7;
  u2 in {r2*x where x is VECTOR of V : x in M2} by A7,CONVEX1:def 1;
  then consider x2 be VECTOR of V such that
A11: u2 = r2*x2 and
A12: x2 in M2;
A13: p*u2 = r2*p*x2 by A11,RLVECT_1:def 7
    .= r2*(p*x2) by RLVECT_1:def 7;
  reconsider r1,r2 as Real;
  p*x2 in M2 by A2,A3,A12;
  then
A14: p*u2 in r2*M2 by A13,RLTOPSP1:1;
  p*x1 in M1 by A1,A3,A9;
  then
A15: p*u1 in r1*M1 by A10,RLTOPSP1:1;
  p*(u1+u2) = p*u1 + p*u2 by RLVECT_1:def 5;
  then
  p*(u1+u2) in {x+y where x,y is VECTOR of V: x in r1*M1 & y in r2*M2} by A15
,A14;
  hence thesis by A5,RUSUB_4:def 9;
end;
