
theorem Th10:
  for X being non empty set, Y be set st Y c= X
  for P being a_partition of X holds
  {a /\ Y where a is Element of P: a meets Y} is a_partition of Y
proof
  let X be non empty set, Y be set;
  assume
A1: Y c= X;
  let P be a_partition of X;
  set Q = {a /\ Y where a is Element of P: a meets Y};
  Q c= bool Y
  proof
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    assume x in Q;
    then ex p being Element of P st ( x = p /\ Y)&( p meets Y);
    then
A2: xx c= X /\ Y by XBOOLE_1:26;
    X /\ Y = Y by A1,XBOOLE_1:28;
    hence thesis by A2;
  end;
  then reconsider Q as Subset-Family of Y;
  Q is a_partition of Y
  proof
    thus union Q c= Y;
    thus Y c= union Q
    proof
      let x be object;
      assume
A3:   x in Y;
      X = union P by EQREL_1:def 4;
      then consider p being set such that
A4:   x in p and
A5:   p in P by A1,A3,TARSKI:def 4;
A6:   p meets Y by A3,A4,XBOOLE_0:3;
A7:   x in p /\ Y by A3,A4,XBOOLE_0:def 4;
      p /\ Y in Q by A5,A6;
      hence thesis by A7,TARSKI:def 4;
    end;
    let A be Subset of Y;
    assume A in Q;
    then consider p being Element of P such that
A8: A = p /\ Y and
A9: p meets Y;
    thus A <> {} by A8,A9;
    let B be Subset of Y;
    assume B in Q;
    then consider p1 being Element of P such that
A10: B = p1 /\ Y and p1 meets Y;
    assume A <> B;
    then p misses p1 by A8,A10,EQREL_1:def 4;
    then A misses p1 by A8,XBOOLE_1:74;
    hence thesis by A10,XBOOLE_1:74;
  end;
  hence thesis;
end;
