reserve X,Y,Z,Z1,Z2,D for set,x,y for object;
reserve SFX,SFY,SFZ for set;
reserve F,G for Subset-Family of D;
reserve P for Subset of D;

theorem
  for X being set for H, J being Subset-Family of X st H c= J holds
  Intersect J c= Intersect H
proof
  let X be set;
  let H, J be Subset-Family of X such that
A1: H c= J;
  let x be object;
  assume
A2: x in Intersect J;
  then for Y be set st Y in H holds x in Y by A1,Th43;
  hence thesis by A2,Th43;
end;
