reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th7:
  for X, Y being TopSpace for V being Subset of X, W being Subset
  of Y holds Int [:V,W:] = [:Int V, Int W:]
proof
  let X, Y be TopSpace, V be Subset of X, W be Subset of Y;
  thus Int [:V,W:] c= [:Int V, Int W:]
  proof
    let e be object;
    assume e in Int [:V,W:];
    then consider Q being Subset of [:X,Y:]such that
A1: Q is open and
A2: Q c= [:V,W:] and
A3: e in Q by TOPS_1:22;
    consider A being Subset-Family of [:X,Y:] such that
A4: Q = union A and
A5: for e st e in A ex X1 being Subset of X, Y1 being Subset of Y st e
    = [:X1,Y1:] & X1 is open & Y1 is open by A1,Th5;
    consider a being set such that
A6: e in a and
A7: a in A by A3,A4,TARSKI:def 4;
    consider X1 being Subset of X, Y1 being Subset of Y such that
A8: a = [:X1,Y1:] and
A9: X1 is open and
A10: Y1 is open by A5,A7;
    [:X1,Y1:] c= Q by A4,A7,A8,ZFMISC_1:74;
    then
A11: [:X1,Y1:] c= [:V,W:] by A2;
    then Y1 c= W by A6,A8,ZFMISC_1:114;
    then
A12: Y1 c= Int W by A10,TOPS_1:24;
    X1 c= V by A6,A8,A11,ZFMISC_1:114;
    then X1 c= Int V by A9,TOPS_1:24;
    then [:X1,Y1:] c= [:Int V, Int W:] by A12,ZFMISC_1:96;
    hence thesis by A6,A8;
  end;
  Int V c= V & Int W c= W by TOPS_1:16;
  then [:Int V, Int W:] c= [:V,W:] by ZFMISC_1:96;
  hence thesis by Th6,TOPS_1:24;
end;
