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

theorem Th24:
  for p being Point of TOP-REAL 2 st p is Point of
  Topen_unit_circle(c[10]) & p`2 = 0 holds p = c[-10]
proof
  let p be Point of TOP-REAL 2 such that
A1: p is Point of Topen_unit_circle(c[10]) and
A2: p`2 = 0;
A3: now
    assume p`1 = 1;
    then p = c[10] by A2,EUCLID:53;
    hence contradiction by A1,Th21;
  end;
  p is Point of TUC by A1,PRE_TOPC:25;
  then 1^2 = |. p .|^2 by Th12
    .= p`1^2+p`2^2 by JGRAPH_3:1;
  then p`1 = 1 or p`1 = -1 by A2,SQUARE_1:41;
  hence thesis by A2,A3,EUCLID:53;
end;
