reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem Th14:
  for f being Function st A c= bool dom f holds f.:(union A) = union((.:f).:A)
proof
  let f be Function such that
A1: A c= bool dom f;
A2: f.:(union A) c= union((.:f).:A)
  proof
    let y be object;
    assume y in f.:(union A);
    then consider x being object such that
    x in dom f and
A3: x in union A and
A4: y = f.x by FUNCT_1:def 6;
    consider X such that
A5: x in X and
A6: X in A by A3,TARSKI:def 4;
    X in bool dom f by A1,A6;
    then X in dom(.:f) by Def1;
    then
A7: (.:f).X in (.:f).:A by A6,FUNCT_1:def 6;
    y in f.:X by A1,A4,A5,A6,FUNCT_1:def 6;
    then y in (.:f).X by A1,A6,Def1;
    hence thesis by A7,TARSKI:def 4;
  end;
  union((.:f).:A) c= f.:(union A) by Th13;
  hence thesis by A2,XBOOLE_0:def 10;
end;
