 reserve n for Nat;

theorem Th33:
   for R being Ring, S being R-monomorphic R-homomorphic Ring
   for h being Monomorphism of R,S
   for p being Element of the carrier of (Polynom-Ring R)
   holds LM((PolyHom h).p) = (PolyHom h).(LM p)
   proof
     let R be Ring, S be R-monomorphic R-homomorphic Ring;
     let h be Monomorphism of R,S;
     let p be Element of the carrier of (Polynom-Ring R);
     reconsider f = (PolyHom h).p as
     Element of the carrier of Polynom-Ring S;
     reconsider LMp = LM p as
     Element of the carrier of Polynom-Ring R by POLYNOM3:def 10;
A1:   deg f = deg p by Th32 .= len p - 1;
     now let i be Element of NAT;
       per cases;
         suppose A2: i = len p -'1; then
           (LM f).i = f.(len p -'1) by A1,POLYNOM4:def 1
                 .= h.(p.(len p -'1)) by Def2
                 .= h.((LM p).i) by A2,POLYNOM4:def 1
                 .= ((PolyHom h).(LMp)).i by Def2;
           hence (LM f).i = ((PolyHom h).(LMp)).i;
         end;
         suppose A3: i <> len p -'1; then
           (LM f).i = 0.S by A1,POLYNOM4:def 1
                 .= h.(0.R) by RING_2:6
                 .= h.((LM p).i) by A3,POLYNOM4:def 1
                 .= ((PolyHom h).(LMp)).i by Def2;
           hence (LM f).i = ((PolyHom h).(LMp)).i;
         end;
       end;
       hence thesis by FUNCT_2:63;
    end;
