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 Th17:
  i in dom f implies |.{f}.|.i = {f.i}
proof
A1: f in {f} by TARSKI:def 1;
  consider A being non empty functional set such that
A2: A = {f} and
  dom |.{f}.| = meet the set of all  dom x where x is Element of A  and
A3: for i st i in dom |.{f}.| holds |.{f}.|.i = the set of all
 x.i where x is Element
  of A  by Def2;
  assume i in dom f;
  then i in dom |.{f}.| by Th15;
  then
A4: |.{f}.|.i = the set of all  x.i where x is Element of A  by A3;
  thus |.{f}.|.i c= { f.i }
  proof
    let q be object;
    assume q in |.{f}.|.i;
    then consider x being Element of A such that
A5: q = x.i by A4;
    x = f by A2,TARSKI:def 1;
    hence thesis by A5,TARSKI:def 1;
  end;
  let q be object;
  assume q in { f.i };
  then q = f.i by TARSKI:def 1;
  hence thesis by A2,A4,A1;
end;
