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

theorem Th16:
  for H being Subset-Family of [:X,Y:], C being set st C in Pr1(X,
  Y).:H ex D being Subset of [:X,Y:] st D in H & C = pr1(the carrier of X, the
  carrier of Y).:D
proof
  let H be Subset-Family of [:X,Y:], C be set;
  assume C in Pr1(X,Y).:H;
  then consider u being object such that
A1: u in dom Pr1(X,Y) and
A2: u in H and
A3: C = Pr1(X,Y).u by FUNCT_1:def 6;
  reconsider u as Subset of [:X,Y:] by A1;
  take u;
  thus u in H by A2;
  the carrier of [:X,Y:] = [:the carrier of X, the carrier of Y:] by Def2;
  hence thesis by A3,EQREL_1:47;
end;
