
theorem Th13:
  for X being set, P being non-empty Function st Union P = X &
  for x,y being set st x in dom P & y in dom P & x <> y holds P.x misses P.y
  holds P is IndexedPartition of X
proof
  let X be set, P be non-empty Function such that
A1: Union P = X and
A2: for x,y being set st x in dom P & y in dom P & x <> y holds P.x misses P.y;
  rng P c= bool X
  proof
    let y be object;
     reconsider yy=y as set by TARSKI:1;
    assume y in rng P;
    then yy c= union rng P by ZFMISC_1:74;
    then yy c= X by A1,CARD_3:def 4;
    hence thesis;
  end;
  then reconsider Y = rng P as Subset-Family of X;
  Y is a_partition of X
  proof
    thus union Y = X by A1,CARD_3:def 4;
    let A be Subset of X;
    assume
A3: A in Y;
    then
A4: ex x being object st x in dom P & A = P.x by FUNCT_1:def 3;
    thus A<>{} by A3;
    let B be Subset of X;
    assume B in Y;
    then ex y being object st y in dom P & B = P.y by FUNCT_1:def 3;
    hence thesis by A2,A4;
  end;
  hence rng P is a_partition of X;
  let x,y be object;
  assume that
A5: x in dom P and
A6: y in dom P and
A7: P.x = P.y and
A8: x <> y;
  reconsider Px = P.x, Py = P.y as non empty set by A5,A6,FUNCT_1:def 9;
  set a = the Element of Px;
  P.x misses P.y by A2,A5,A6,A8;
  then not a in Py by XBOOLE_0:3;
  hence contradiction by A7;
end;
