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 regular
proof
  let p;
  let P be Subset of the_Complex_Space n such that
A1: P <> {} and
A2: P is closed & p in P`;
  reconsider A = P as Subset of COMPLEX n;
  reconsider z1 = p as Element of COMPLEX n;
  set d = dist(z1,A)/2;
  reconsider K1 = Ball(z1,d), K2 = Ball(A,d) as Subset of the_Complex_Space n;
  take K1,K2;
A3: Ball(z1,d) is open by SEQ_4:112;
  Ball(A,d) is open by A1,SEQ_4:122;
  hence K1 is open & K2 is open by A3;
  A is closed & not p in P by A2,Th5,XBOOLE_0:def 5;
  then 0 < d by A1,SEQ_4:117,XREAL_1:215;
  hence p in K1 & P c= K2 by SEQ_4:111,121;
  assume K1 /\ K2 <> {};
  then consider x such that
A4: x in Ball(z1,d) /\ Ball(A,d) by SUBSET_1:4;
  x in K2 by A4,XBOOLE_0:def 4;
  then
A5: dist(x,A) < d by SEQ_4:119;
  x in K1 by A4,XBOOLE_0:def 4;
  then |.z1 - x.| < d by SEQ_4:110;
  then |.z1 - x.| + dist(x,A) < d + d by A5,XREAL_1:8;
  hence contradiction by A1,SEQ_4:118;
end;
