 reserve n for Nat;

theorem
   for R being non degenerated Ring, S being R-homomorphic Ring
   for h being Homomorphism of R,S
   for p being non zero Element of the carrier of (Polynom-Ring R)
   holds deg((PolyHom h).p) = deg p iff h.(LC p) <> 0.S
   proof
     let R be non degenerated Ring, S be R-homomorphic Ring;
     let h be Homomorphism of R,S;
     let p be non zero Element of the carrier of (Polynom-Ring R);
     p <> 0_.R; then
A1:   len p <> 0 by POLYNOM4:5;
     hereby
     assume deg((PolyHom h).p) = deg p; then
     h.(p.(len p-'1)) <> 0.S by A1,Lm3;
     hence h.(LC p) <> 0.S by RATFUNC1:def 6;
     end;
     assume h.(LC p) <> 0.S; then
     h.(p.(len p-'1)) <> 0.S by RATFUNC1:def 6;
     hence deg((PolyHom h).p) = deg p by A1,Lm3;
   end;
