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
  for F being Function of I,the carrier' of C holds f*F = (I-->f)"*"F
proof
  let F be Function of I,the carrier' of C;
  now
    let x;
    assume
A1: x in I;
    hence (f*F)/.x = f(*)(F/.x) by Def6
      .= ((I-->f)/.x)(*)(F/.x) by A1,Th2
      .= ((I-->f)"*"F)/.x by A1,Def7;
  end;
  hence thesis by Th1;
end;
