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

theorem Th13:
  for x being Point of TOP-REAL 2 st x is Point of Tunit_circle(2)
  holds -1 <= x`1 & x`1 <= 1 & -1 <= x`2 & x`2 <= 1
proof
  let x be Point of TOP-REAL 2 such that
A1: x is Point of Tunit_circle(2);
  consider a, b being Element of REAL such that
A2: x = <*a,b*> by EUCLID:51;
A3: b = x`2 by A2,EUCLID:52;
A4: a = x`1 by A2,EUCLID:52;
A5: 1^2 = |. x .|^2 by A1,Th12
    .= a^2+b^2 by A4,A3,JGRAPH_3:1;
  0 <= a^2 by XREAL_1:63;
  then -a^2 <= -0;
  then
A6: b^2-1 <= 0 by A5;
  0 <= b^2 by XREAL_1:63;
  then -b^2 <= -0;
  then a^2-1 <= 0 by A5;
  hence thesis by A4,A3,A6,SQUARE_1:43;
end;
