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 Th17:
  for F being Function of I,the carrier' of C st cods F = I-->dom
  f holds doms(f*F) = doms F & cods(f*F) = I-->cod f
proof
  let F be Function of I,the carrier' of C such that
A1: cods F = I-->dom f;
  now
    let x;
    assume
A2: x in I;
    then
A3: cod(F/.x) = (I-->dom f)/.x by A1,Def2
      .= dom f by A2,Th2;
    thus (doms F)/.x = dom(F/.x) by A2,Def1
      .= dom(f(*)(F/.x)) by A3,CAT_1:17
      .= dom((f*F)/.x) by A2,Def6
      .= (doms(f*F))/.x by A2,Def1;
  end;
  hence doms(f*F) = doms F by Th1;
  now
    let x;
    assume
A4: x in I;
    then
A5: cod(F/.x) = (I-->dom f)/.x by A1,Def2
      .= dom f by A4,Th2;
    thus (cods(f*F))/.x = cod((f*F)/.x) by A4,Def2
      .= cod(f(*)(F/.x)) by A4,Def6
      .= cod f by A5,CAT_1:17
      .= (I--> cod f)/.x by A4,Th2;
  end;
  hence thesis by Th1;
end;
