reserve I for set,
  x,x1,x2,y,z for set,
  A for non empty set;
reserve C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,g,h,i,j,k,p1,p2,q1,q2,i1,i2,j1,j2 for Morphism of C;

theorem Th16:
  for F being Function of I,the carrier' of C st doms F = I-->cod
  f holds doms(F*f) = I-->dom f & cods(F*f) = cods F
proof
  let F be Function of I,the carrier' of C such that
A1: doms F = I-->cod f;
  now
    let x;
    assume
A2: x in I;
    then
A3: dom(F/.x) = (I-->cod f)/.x by A1,Def1
      .= cod f by A2,Th2;
    thus (doms(F*f))/.x = dom((F*f)/.x) by A2,Def1
      .= dom((F/.x)(*)f) by A2,Def5
      .= dom f by A3,CAT_1:17
      .= (I--> dom f)/.x by A2,Th2;
  end;
  hence doms(F*f) = I --> dom f by Th1;
  now
    let x;
    assume
A4: x in I;
    then
A5: dom(F/.x) = (I-->cod f)/.x by A1,Def1
      .= cod f by A4,Th2;
    thus (cods F)/.x = cod(F/.x) by A4,Def2
      .= cod((F/.x)(*)f) by A5,CAT_1:17
      .= cod((F*f)/.x) by A4,Def5
      .= (cods(F*f))/.x by A4,Def2;
  end;
  hence thesis by Th1;
end;
