
theorem Th11:
  for f being non empty Function, P being a_partition of dom f holds
  the set of all f|a where a is Element of P is a_partition of f
proof
  let f be non empty Function;
  set X = dom f;
  let P be a_partition of X;
  set Y = f;
  set Q = the set of all f|a where a is Element of P;
  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 = f|p);
    then xx c= f by RELAT_1:59;
    hence thesis;
  end;
  then reconsider Q as Subset-Family of Y;
  Q is a_partition of Y
  proof
    Y c= union Q
    proof
      let y,z be object;
      assume
A1:   [y,z] in f;
      then
A2:   y in X by XTUPLE_0:def 12;
      X = union P by EQREL_1:def 4;
      then consider p being set such that
A3:   y in p and
A4:   p in P by A2,TARSKI:def 4;
A5:   [y,z] in f|p by A1,A3,RELAT_1:def 11;
      f|p in Q by A4;
      hence thesis by A5,TARSKI:def 4;
    end;
    hence Y = union Q;
    let A be Subset of Y;
    assume A in Q;
    then consider p being Element of P such that
A6: A = f|p;
    reconsider p as non empty Subset of X;

    thus A <> {} by A6;
    let B be Subset of Y;
    assume B in Q;
    then consider p1 being Element of P such that
A7: B = f|p1;
    assume A <> B;
    then
A8: p misses p1 by A6,A7,EQREL_1:def 4;
    assume A meets B;
    then consider x being object such that
A9: x in A and
A10: x in B by XBOOLE_0:3;
    consider y,z being object such that
A11: x = [y,z] by A9,RELAT_1:def 1;
A12: y in p by A6,A9,A11,RELAT_1:def 11;
    y in p1 by A7,A10,A11,RELAT_1:def 11;
    hence contradiction by A8,A12,XBOOLE_0:3;
  end;
  hence thesis;
end;
