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

theorem Th27:
  for x being Point of TOP-REAL 2 st x is Point of
  Topen_unit_circle(c[10]) holds -1 <= x`1 & x`1 < 1
proof
  let x be Point of TOP-REAL 2;
  assume
A1: x is Point of Topen_unit_circle(c[10]);
A2: now
A3: the carrier of TOUC = cS1 \ {c[10]} by Def10;
    then
A4: not x in {c[10]} by A1,XBOOLE_0:def 5;
A5: x = |[x`1,x`2]| by EUCLID:53;
    assume
A6: x`1 = 1;
    x in cS1 by A1,A3,XBOOLE_0:def 5;
    then x = c[10] by A6,A5,Th14;
    hence contradiction by A4,TARSKI:def 1;
  end;
  x`1 <= 1 by A1,Th26;
  hence thesis by A1,A2,Th26,XXREAL_0:1;
end;
