reserve n for Element of NAT,
  i for Integer,
  a, b, r for Real,
  x for Point of TOP-REAL n;

theorem Th10:
  Tcircle(0.TOP-REAL 2,r) is SubSpace of Trectangle(-r,r,-r,r)
proof
  set C = Tcircle(0.TOP-REAL 2,r);
  set T = Trectangle(-r,r,-r,r);
  the carrier of C c= the carrier of T
  proof
    let x be object;
A1: closed_inside_of_rectangle(-r,r,-r,r) = {p where p is Point of
    TOP-REAL 2: -r <= p`1 & p`1 <= r & -r <= p`2 & p`2 <= r} by JGRAPH_6:def 2;
    assume
A2: x in the carrier of C;
    reconsider x as Point of TOP-REAL 2 by A2,PRE_TOPC:25;
    the carrier of C = Sphere(0.TOP-REAL 2,r) by Th9;
    then
A3: |. x .| = r by A2,TOPREAL9:12;
A4: |.x`2.| <= |. x .| by JGRAPH_1:33;
    then
A5: -r <= x`2 by A3,ABSVALUE:5;
A6: |.x`1.| <= |. x .| by JGRAPH_1:33;
    then
A7: x`1 <= r by A3,ABSVALUE:5;
A8: the carrier of Trectangle(-r,r,-r,r) = closed_inside_of_rectangle(-r,r
    ,-r,r) by PRE_TOPC:8;
A9: x`2 <= r by A3,A4,ABSVALUE:5;
    -r <= x`1 by A3,A6,ABSVALUE:5;
    hence thesis by A1,A8,A7,A5,A9;
  end;
  hence thesis by TSEP_1:4;
end;
