
theorem :: Ex1:
  for X being set, K being Subset of X holds { [A, B] where A, B is
  Subset of X : K c= A or B c= A } is Full-family of X
proof
  let X be set, K be Subset of X;
  set F = { [A, B] where A, B is Subset of X : K c= A or B c= A };
  F c= [:bool X, bool X:]
  proof
    let x be object;
    assume x in F;
    then ex A, B being Subset of X st x = [A, B] & (K c= A or B c= A);
    hence thesis;
  end;
  then reconsider F as Dependency-set of X;
A1: F is (F4)
  proof
    let A, B, A9, B9 be Subset of X;
    assume that
A2: [A, B] in F and
A3: [A9, B9] in F;
    consider a, b being Subset of X such that
A4: [A, B] = [a, b] and
A5: K c= a or b c= a by A2;
A6: B = b by A4,XTUPLE_0:1;
    consider a9, b9 being Subset of X such that
A7: [A9, B9] = [a9, b9] and
A8: K c= a9 or b9 c= a9 by A3;
A9: A9 = a9 by A7,XTUPLE_0:1;
A10: B9 = b9 by A7,XTUPLE_0:1;
    A = a by A4,XTUPLE_0:1;
    then K c= A\/A9 or B\/B9 c= A\/A9 by A5,A8,A6,A9,A10,XBOOLE_1:10,13;
    hence thesis;
  end;
  now
    let A, B, C be Subset of X;
    assume that
A11: [A, B] in F and
A12: [B, C] in F;
    consider a, b being Subset of X such that
A13: [A, B] = [a, b] and
A14: K c= a or b c= a by A11;
A15: A = a by A13,XTUPLE_0:1;
    consider b1, c being Subset of X such that
A16: [B, C] = [b1, c] and
A17: K c= b1 or c c= b1 by A12;
A18: B = b1 by A16,XTUPLE_0:1;
A19: C = c by A16,XTUPLE_0:1;
    B = b by A13,XTUPLE_0:1;
    then K c= a or c c= a by A14,A17,A18;
    hence [A, C] in F by A15,A19;
  end;
  then
A20: F is (F2) by Th18;
  F is (DC3);
  hence thesis by A20,A1;
end;
