reserve i, x, I for set,
  A, B, M for ManySortedSet of I,
  f, f1 for Function;
reserve SF, SG for SubsetFamily of M;
reserve E, T for Element of Bool M;

theorem Th15:
  dom |.{f}.| = dom f
proof
  consider A being non empty functional set such that
A1: A = {f} and
A2: dom |.{f}.| = meet the set of all  dom x where x is Element of A  and
  for i st i in dom |.{f}.| holds |.{f}.|.i = the set of all
 x.i where x is Element of
A by Def2;
  set m = the set of all  dom x where x is Element of A ;
  m = {dom f}
  proof
    thus m c= {dom f}
    proof
      let q be object;
      assume q in m;
      then consider x being Element of A such that
A3:   q = dom x;
      x = f by A1,TARSKI:def 1;
      hence thesis by A3,TARSKI:def 1;
    end;
    let q be object;
    assume q in {dom f};
    then
A4: q = dom f by TARSKI:def 1;
    f is Element of A by A1,TARSKI:def 1;
    hence thesis by A4;
  end;
  hence thesis by A2,SETFAM_1:10;
end;
