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

theorem Th15:
  for A being Subset of [:X,Y:], H being Subset-Family of [:X,Y:]
st for e st e in H holds e c= A & ex X1 being Subset of X, Y1 being Subset of Y
  st e =[:X1,Y1:] holds [:union(Pr1(X,Y).:H), meet(Pr2(X,Y).:H):] c= A
proof
  let A be Subset of [:X,Y:], H be Subset-Family of [:X,Y:];
  assume
A1: for e st e in H holds e c= A & ex X1 being Subset of X, Y1 being
  Subset of Y st e =[:X1,Y1:];
  the carrier of [:X,Y:] = [:the carrier of X, the carrier of Y:] by Def2;
  hence thesis by A1,EQREL_1:51;
end;
