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

theorem Th21:
  for H being Subset-Family of [:X,Y:], X1 being Subset of X, Y1
being Subset of Y st H is Cover of [:X1,Y1:] holds (Y1 <> {} implies Pr1(X,Y).:
  H is Cover of X1) & (X1 <> {} implies Pr2(X,Y).:H is Cover of Y1)
proof
  let H be Subset-Family of [:X,Y:], X1 be Subset of X, Y1 be Subset of Y;
A1: dom .:pr2(the carrier of X, the carrier of Y) = bool dom pr2(the
  carrier of X, the carrier of Y) by FUNCT_3:def 1
    .= bool[:the carrier of X,the carrier of Y:] by FUNCT_3:def 5;
A2: the carrier of [:X,Y:] = [:the carrier of X, the carrier of Y:] by Def2;
  assume
A3: [:X1,Y1:] c= union H;
  thus Y1 <> {} implies Pr1(X,Y).:H is Cover of X1
  proof
    assume Y1 <> {};
    then consider y being Point of Y such that
A4: y in Y1 by SUBSET_1:4;
    let e be object;
    assume
A5: e in X1;
    then reconsider x = e as Point of X;
    [x,y] in [:X1,Y1:] by A4,A5,ZFMISC_1:87;
    then consider A being set such that
A6: [x,y] in A and
A7: A in H by A3,TARSKI:def 4;
    [x,y] in [:the carrier of X, the carrier of Y:] by ZFMISC_1:87;
    then
A8: [x,y] in dom pr1(the carrier of X, the carrier of Y) by FUNCT_3:def 4;
A9: dom .:pr1(the carrier of X, the carrier of Y) = bool dom pr1(the
    carrier of X, the carrier of Y) by FUNCT_3:def 1
      .= bool[:the carrier of X,the carrier of Y:] by FUNCT_3:def 4;
    e = pr1(the carrier of X, the carrier of Y).(x,y) by FUNCT_3:def 4;
    then
A10: e in pr1(the carrier of X, the carrier of Y).:A by A6,A8,FUNCT_1:def 6;
    .:pr1(the carrier of X, the carrier of Y).A = pr1(the carrier of X,
    the carrier of Y).:A by A2,A7,EQREL_1:47;
    then
    pr1(the carrier of X, the carrier of Y).:A in Pr1(X,Y).:H by A2,A7,A9,
FUNCT_1:def 6;
    hence e in union (Pr1(X,Y).:H) by A10,TARSKI:def 4;
  end;
  assume X1 <> {};
  then consider x being Point of X such that
A11: x in X1 by SUBSET_1:4;
  let e be object;
  assume
A12: e in Y1;
  then reconsider y = e as Point of Y;
  [x,y] in [:X1,Y1:] by A11,A12,ZFMISC_1:87;
  then consider A being set such that
A13: [x,y] in A and
A14: A in H by A3,TARSKI:def 4;
  [x,y] in [:the carrier of X, the carrier of Y:] by ZFMISC_1:87;
  then
A15: [x,y] in dom pr2(the carrier of X, the carrier of Y) by FUNCT_3:def 5;
  e = pr2(the carrier of X, the carrier of Y).(x,y) by FUNCT_3:def 5;
  then
A16: e in pr2(the carrier of X, the carrier of Y).:A by A13,A15,FUNCT_1:def 6;
  .:pr2(the carrier of X, the carrier of Y).A = pr2(the carrier of X, the
  carrier of Y).:A by A2,A14,EQREL_1:48;
  then
  pr2(the carrier of X, the carrier of Y).:A in Pr2(X,Y).:H by A2,A14,A1,
FUNCT_1:def 6;
  hence e in union (Pr2(X,Y).:H) by A16,TARSKI:def 4;
end;
