
theorem
  for Y0, Y1 being TopStruct, D0 being Subset of Y0, D1 being Subset of
  Y1 st the TopStruct of Y0 = the TopStruct of Y1 & D0 = D1 holds D0 is T_0
  implies D1 is T_0
proof
  let Y0, Y1 be TopStruct, D0 be Subset of Y0, D1 be Subset of Y1;
  assume
A1: the TopStruct of Y0 = the TopStruct of Y1;
  assume
A2: D0 = D1;
  assume
A3: D0 is T_0;
  now
    let x, y be Point of Y1;
    reconsider x0 = x, y0 = y as Point of Y0 by A1;
    assume
A4: x in D1 & y in D1;
    assume
A5: x <> y;
    hereby
      per cases by A2,A3,A4,A5;
      suppose
        ex V being Subset of Y0 st V is open & x0 in V & not y0 in V;
        then consider V being Subset of Y0 such that
A6:     V is open and
A7:     x0 in V & not y0 in V;
        reconsider V1 = V as Subset of Y1 by A1;
        now
          take V1;
          V1 in the topology of Y1 by A1,A6,PRE_TOPC:def 2;
          hence V1 is open by PRE_TOPC:def 2;
          thus x in V1 & not y in V1 by A7;
        end;
        hence (ex V1 being Subset of Y1 st V1 is open & x in V1 & not y in V1)
        or ex W1 being Subset of Y1 st W1 is open & not x in W1 & y in W1;
      end;
      suppose
        ex W being Subset of Y0 st W is open & not x0 in W & y0 in W;
        then consider W being Subset of Y0 such that
A8:     W is open and
A9:     ( not x0 in W)& y0 in W;
        reconsider W1 = W as Subset of Y1 by A1;
        now
          take W1;
          W1 in the topology of Y1 by A1,A8,PRE_TOPC:def 2;
          hence W1 is open by PRE_TOPC:def 2;
          thus not x in W1 & y in W1 by A9;
        end;
        hence (ex V1 being Subset of Y1 st V1 is open & x in V1 & not y in V1)
        or ex W1 being Subset of Y1 st W1 is open & not x in W1 & y in W1;
      end;
    end;
  end;
  hence thesis;
end;
