 reserve n for Nat;

theorem Th32:
   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 deg((PolyHom h).p) = deg 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;
A1:   now let i be Nat;
       assume i >= len p; then
       p.i = 0.R by ALGSEQ_1:8;
       hence f.i = h.(0.R) by Def2 .= 0.S by RING_2:6;
     end;
     now let m be Nat;
       assume
A2:     m is_at_least_length_of f;
       now assume len p > m; then
         len p - 1 > m - 1 by XREAL_1:6; then
A3:      len p - 1 >= (m - 1) + 1 by INT_1:7; then
         reconsider lp = len p - 1 as Element of NAT by INT_1:3;
A4:      lp + 1 = len p;
         h.(0.R) = 0.S by RING_2:6
           .= f.lp by A3,A2
           .= h.(p.lp) by Def2; then
         p.lp = 0.R by FUNCT_2:19;
         hence contradiction by A4,ALGSEQ_1:10;
       end;
       hence len p <= m;
     end;
     hence thesis by A1,ALGSEQ_1:def 2,def 3;
   end;
