
theorem Th1:
  for V being RealLinearSpace, A,B being circled Subset of V holds
  A-B is circled
proof
  let V be RealLinearSpace,A,B be circled Subset of V;
A1: A-B = {u - v where u,v is VECTOR of V: u in A & v in B} by RUSUB_5:def 2;
  let r be Real;
  assume |.r.| <= 1;
  then
A2: r*A c= A & r*B c= B by RLTOPSP1:def 6;
  let x be object;
  assume
A3: x in r*(A-B);
  r*(A-B) = {r * v where v is VECTOR of V: v in A - B} by CONVEX1:def 1;
  then consider x9 being VECTOR of V such that
A4: x = r*x9 and
A5: x9 in A-B by A3;
  consider u,v being VECTOR of V such that
A6: x9 = u-v and
A7: u in A & v in B by A1,A5;
  reconsider r as Real;
A8: r*u in r*A & r*v in r*B by A7,RLTOPSP1:1;
  x = r*u-r*v by A4,A6,RLVECT_1:34;
  hence thesis by A2,A8,Lm1;
end;
