
theorem Th25:
  for X being non empty TopSpace for Y being non trivial
  T_0-TopSpace st oContMaps(X, Y) is with_suprema holds not Y is T_1
proof
  let X be non empty TopSpace;
  let Y be non trivial T_0-TopSpace;
  consider a,b being Element of Y such that
A1: a <> b by STRUCT_0:def 10;
  set i = the Element of X;
  reconsider f = X --> a, g = X --> b as continuous Function of X, Y;
  assume oContMaps(X, Y) is with_suprema;
  then reconsider XY = oContMaps(X, Y) as sup-Semilattice;
  reconsider ef = f, eg = g as Element of XY by Th2;
  reconsider h = ef "\/" eg, f = ef, g = eg as continuous Function of X, Omega
  Y by Th1;
A2: f.i = a & g.i = b by FUNCOP_1:7;
  now
    eg <= ef "\/" eg by YELLOW_0:22;
    then g <= h by Th3;
    then
A3: ex x,y being Element of Omega Y st x = g.i & y = h.i & x <= y;
    ef <= ef "\/" eg by YELLOW_0:22;
    then f <= h by Th3;
    then
A4: ex x,y being Element of Omega Y st x = f.i & y = h.i & x <= y;
    assume
A5: not ex x,y being Element of Omega Y st x <= y & x <> y;
    then not (f.i <= h.i & f.i <> h.i);
    hence contradiction by A1,A2,A5,A4,A3;
  end;
  then consider x,y being Element of Omega Y such that
A6: x <= y and
A7: x <> y;
A8: the TopStruct of Omega Y = the TopStruct of Y by WAYBEL25:def 2;
  then reconsider p = x, q = y as Element of Y;
  take p,q;
  thus p <> q by A7;
  let W,V be Subset of Y;
  assume W is open;
  then reconsider W as open Subset of Omega Y by A8,TOPS_3:76;
  W is upper;
  hence thesis by A6;
end;
