reserve
  r,s,r0,s0,t for Real;

theorem Th4:
  for S,T being non empty TopSpace, G being Subset of [:S,T:] st
for x being Point of [:S,T:] st x in G ex GS being Subset of S, GT being Subset
  of T st GS is open & GT is open & x in [:GS,GT:] & [:GS,GT:] c= G holds G is
  open
proof
  let S,T be non empty TopSpace, G being Subset of [:S,T:] such that
A1: for x being Point of [:S,T:] st x in G ex GS being Subset of S, GT
being Subset of T st GS is open & GT is open & x in [:GS,GT:] & [:GS,GT:] c= G;
  set A = {[:GS,GT:] where GS is Subset of S, GT is Subset of T : GS is open &
  GT is open & [:GS,GT:] c= G };
  A c= bool the carrier of [:S,T:]
  proof
    let e be object;
    assume e in A;
    then
    ex GS being Subset of S, GT being Subset of T st e = [:GS, GT:] & GS
    is open & GT is open & [:GS,GT:] c= G;
    hence thesis;
  end;
  then reconsider A as Subset-Family of [:S,T:];
  reconsider A as Subset-Family of [:S,T:];
  for x being object holds x in G iff ex Y being set st x in Y & Y in A
  proof
    let x be object;
    thus x in G implies ex Y being set st x in Y & Y in A
    proof
      assume x in G;
      then consider GS being Subset of S, GT being Subset of T such that
A2:   GS is open and
A3:   GT is open and
A4:   x in [:GS,GT:] and
A5:   [:GS,GT:] c= G by A1;
      take [:GS,GT:];
      thus thesis by A2,A3,A4,A5;
    end;
    given Y being set such that
A6: x in Y and
A7: Y in A;
    ex GS being Subset of S, GT being Subset of T st Y = [:GS,GT:] & GS
    is open & GT is open & [:GS,GT:] c= G by A7;
    hence thesis by A6;
  end;
  then
A8: G = union A by TARSKI:def 4;
  for e being set st e in A ex X1 being Subset of S, Y1 being Subset of T
  st e = [:X1,Y1:] & X1 is open & Y1 is open
  proof
    let e be set;
    assume e in A;
    then ex GS being Subset of S, GT being Subset of T st e = [:GS,GT:] & GS
    is open & GT is open & [:GS,GT:] c= G;
    hence thesis;
  end;
  hence thesis by A8,BORSUK_1:5;
end;
