reserve n for Element of NAT,
        x for Element of COMPLEX n;
reserve p,q for Point of the_Complex_Space n,
  V for Subset of the_Complex_Space n;

theorem
  the_Complex_Space n is T_2
proof
  let p,q;
  assume
A1: p <> q;
  reconsider z1 = p, z2 = q as Element of COMPLEX n;
  set d = |. z1 - z2 .|/2;
  reconsider K1 = Ball(z1,d), K2 = Ball(z2,d) as Subset of the_Complex_Space n;
  take K1,K2;
  Ball(z1,d) is open & Ball(z2,d) is open by SEQ_4:112;
  hence K1 is open & K2 is open;
  0 < |. z1 - z2 .| by A1,SEQ_4:103;
  hence p in K1 & q in K2 by SEQ_4:111,XREAL_1:215;
  assume K1 /\ K2 <> {};
  then consider x such that
A2: x in Ball(z1,d) /\ Ball(z2,d) by SUBSET_1:4;
  x in K2 by A2,XBOOLE_0:def 4;
  then
A3: |.z2 - x .| < d by SEQ_4:110;
  x in K1 by A2,XBOOLE_0:def 4;
  then |.z1 - x.| < d by SEQ_4:110;
  then |.z1 - x.| + |.z2 - x.| < d + d by A3,XREAL_1:8;
  then |.z1 - x.| + |.x - z2.| < |.z1 - z2.| by SEQ_4:104;
  hence contradiction by SEQ_4:105;
end;
