theorem Th44:
  filter_image(s,<. Frechet_Filter(NAT),Frechet_Filter(NAT).)) =
   {M where M is Subset of the carrier of T: ex n being Nat st
   square-uparrow n c= s"(M)}
  proof
    set X = {M where M is Subset of the carrier of T:
      s"(M) in <. Frechet_Filter(NAT),Frechet_Filter(NAT).)},
    Y = {M where M is Subset of the carrier of T:
      ex n being Nat st square-uparrow n c= s"(M)};
    X = Y
    proof
      now
        let x be object;
        assume x in X;
        then consider M be Subset of the carrier of T such that
A1:     x = M and
A2:     s"(M) in <. Frechet_Filter(NAT),Frechet_Filter(NAT).);
        ex n st square-uparrow n c= s"(M) by Th42,A2;
        hence x in Y by A1;
      end; then
A3:   X c= Y;
      now
        let x be object;
        assume x in Y;
        then consider M be Subset of the carrier of T such that
A4:     x = M and
A5:     ex n st square-uparrow n c= s"(M);
        s"(M) in <. Frechet_Filter(NAT),Frechet_Filter(NAT).) by A5,Th42;
        hence x in X by A4;
      end;
      then Y c= X;
      hence thesis by A3;
    end;
    hence thesis;
  end;
