reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];
reserve X,Y,X1,X2 for non empty set,
          cA1,cB1 for filter_base of X1,
          cA2,cB2 for filter_base of X2,
              cF1 for Filter of X1,
              cF2 for Filter of X2,
             cBa1 for basis of cF1,
             cBa2 for basis of cF2;
reserve T for non empty TopSpace,
        s for Function of [:NAT,NAT:], the carrier of T,
        M for Subset of the carrier of T;
reserve cF3,cF4 for Filter of the carrier of T;

theorem Th40:
  for f being Function of X,Y, cFXa,cFXb being Filter of X st
  cFXb is_filter-finer_than cFXa holds
  filter_image(f,cFXb) is_filter-finer_than filter_image(f,cFXa)
  proof
    let f be Function of X,Y, cFXa,cFXb be Filter of X;
    assume
A1: cFXb is_filter-finer_than cFXa;
    filter_image(f,cFXa) c= filter_image(f,cFXb)
    proof
      let x be object;
      assume x in filter_image(f,cFXa);
      then ex M be Subset of Y st x = M & f"(M) in cFXa;
      hence thesis by A1;
    end;
    hence thesis;
  end;
