 reserve K,F,E for Field,
         R,S for Ring;

theorem
   for S being RingExtension of R,
   p being Element of the carrier of Polynom-Ring R,
   q being Element of the carrier of Polynom-Ring S st p = q
   holds deg p = deg q
   proof
     let S be RingExtension of R;
     let p be Element of the carrier of Polynom-Ring R,
     q be Element of the carrier of Polynom-Ring S;
     assume
A1:  p = q;
A2:  R is Subring of S by Def1;
     per cases;
       suppose q is zero; then
         len q = 0 by UPROOTS:17; then
A3:      deg q = 0 - 1 by HURWITZ:def 2; then
         q = 0_.(S) by HURWITZ:20
         .= 0.(Polynom-Ring S) by POLYNOM3:def 10
         .= 0.(Polynom-Ring R) by Th7
         .= 0_.(R) by POLYNOM3:def 10;
         hence deg p = deg q by A1,A3,HURWITZ:20;
       end;
       suppose
A4:      q is non zero; then
         len q > 0 by UPROOTS:17; then
A5:      len q -' 1 = len q - 1 by XREAL_0:def 2; then
         reconsider lenq = len q - 1 as Element of NAT;
A6:      now let i be Nat;
           assume i >= len q; then
           q.i = 0.S by ALGSEQ_1:8;
           hence p.i = 0.R by A1,A2,C0SP1:def 3;
         end;
         now let m be Nat;
           assume
A7:        m is_at_least_length_of p;
           now assume len q > m; then
           lenq + 1 > m; then
           lenq >= m by NAT_1:13; then
           p.(len q-'1) = 0.R by A5,A7,ALGSEQ_1:def 2; then
A8:        q.(len q-'1) = 0.S by A1,A2,C0SP1:def 3;
           0 + 1 < len q + 1 by A4,XREAL_1:8,UPROOTS:17; then
           len q >= 1 by NAT_1:13; then
           (len q-'1) + 1 = len q by XREAL_1:235;
           hence contradiction by A8,ALGSEQ_1:10;
         end;
         hence len q <= m;
       end; then
       len p = len q by A6,ALGSEQ_1:def 3,ALGSEQ_1:def 2;
       hence deg p = len q - 1 by HURWITZ:def 2 .= deg q by HURWITZ:def 2;
     end;
   end;
