reserve r,s,t,u for Real;

theorem Th28:
  for X being RealLinearSpace, A being circled Subset of X for r
  being Real st |.r.| = 1 holds r*A = A
proof
  let X be RealLinearSpace, A be circled Subset of X;
  let r be Real;
  assume
A1: |.r.| = 1;
  hence
A2: r*A c= A by Def6;
  let x be object;
  assume
A3: x in A;
  then reconsider x as Point of X;
A4: r*x in r*A by A3;
  per cases;
  suppose
    0 <= r;
    then r = 1 by A1,ABSVALUE:def 1;
    hence thesis by A3,CONVEX1:32;
  end;
  suppose
    r < 0;
    then |.r.| = -r by ABSVALUE:def 1;
    then (-1)*((-1)*x) in r*A by A1,A2,A4;
    then (-1)*(-1)*x in r*A by RLVECT_1:def 7;
    hence thesis by RLVECT_1:def 8;
  end;
end;
