
theorem
  for X being set, A,B being Subset-Family of X st A = B \/ {X} or B = A \ { X
  } holds Intersect A = Intersect B
proof
  let X be set, A,B be Subset-Family of X;
  assume
A1: A = B \/ {X} or B = A \ {X};
  hereby
    let x be object;
    assume
A2: x in Intersect A;
    now
      let y be set;
      assume y in B;
      then y in A by A1,XBOOLE_0:def 3,def 5;
      hence x in y by A2,SETFAM_1:43;
    end;
    hence x in Intersect B by A2,SETFAM_1:43;
  end;
  let x be object;
  assume
A3: x in Intersect B;
  now
    let y be set;
    assume y in A;
    then y in B & not y in {X} or y in {X} by A1,XBOOLE_0:def 3,def 5;
    then y in B or y = X by TARSKI:def 1;
    hence x in y by A3,SETFAM_1:43;
  end;
  hence thesis by A3,SETFAM_1:43;
end;
