reserve X,Y,Z for set, x,y,z for object;
reserve i,j for Nat;
reserve A,B,C for Subset of X;
reserve R,R1,R2 for Relation of X;
reserve AX for Subset of [:X,X:];
reserve SFXX for Subset-Family of [:X,X:];
reserve EqR,EqR1,EqR2,EqR3 for Equivalence_Relation of X;
reserve X for non empty set,
  x for Element of X;
reserve F for Part-Family of X;
reserve e,u,v for object, E,X,Y,X1 for set;

theorem
  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:] such that
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:];
  let u,v be object;
  assume
A2: [u,v] in [:union(.:pr1(X,Y).:H), meet(.:pr2(X,Y).:H):];
  then
A3: v in meet(.:pr2(X,Y).:H) by ZFMISC_1:87;
  u in union(.:pr1(X,Y).:H) by A2,ZFMISC_1:87;
  then consider x being set such that
A4: u in x and
A5: x in .:pr1(X,Y).:H by TARSKI:def 4;
  consider y being object such that
  y in dom .:pr1(X,Y) and
A6: y in H and
A7: x = .:pr1(X,Y).y by A5,FUNCT_1:def 6;
   reconsider y as set by TARSKI:1;
  consider X1 being Subset of X, Y1 being Subset of Y such that
A8: y =[:X1,Y1:] by A1,A6;
A9: y <> {} by A4,A7,FUNCT_3:8;
  y in bool[:X,Y:] by A6;
  then y in bool dom pr2(X,Y) by FUNCT_3:def 5;
  then y in dom .:pr2(X,Y) by FUNCT_3:def 1;
  then .:pr2(X,Y).y in .:pr2(X,Y).:H by A6,FUNCT_1:def 6;
  then Y1 in .:pr2(X,Y).:H by A8,A9,Th50;
  then
A10: v in Y1 by A3,SETFAM_1:def 1;
  u in X1 by A4,A7,A8,A9,Th50;
  then
A11: [u,v] in y by A8,A10,ZFMISC_1:87;
  y c= A by A1,A6;
  hence thesis by A11;
end;
