reserve i, j, k, c, m, n for Nat,
  a, x, y, z, X, Y for set,
  D, E for non empty set,
  R for Relation,
  f, g for Function,
  p, q for FinSequence;

theorem Th12:
  for X being functional compatible set, f being Function st f in X holds
  dom f c= dom union X & for x being set st x in dom f holds (union X).x = f.x
proof
  let X be functional compatible set, f be Function such that
A1: f in X;
  thus dom f c= dom union X
  proof
    let x be object;
    assume x in dom f;
    then consider y being object such that
A2: [x,y] in f by XTUPLE_0:def 12;
    [x, y] in union X by A1,A2,TARSKI:def 4;
    hence thesis by XTUPLE_0:def 12;
  end;
  let x being set;
  assume x in dom f;
  then consider y being object such that
A3: [x,y] in f by XTUPLE_0:def 12;
  [x, y] in union X by A1,A3,TARSKI:def 4;
  hence (union X).x = y by FUNCT_1:1
    .= f.x by A3,FUNCT_1:1;
end;
