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

theorem
   for R being non degenerated Ring, S being RingExtension of R,
   a being Element of R, b being Element of S st a = b
   holds rpoly(1,a) = rpoly(1,b)
   proof
     let R be non degenerated Ring, S be RingExtension of R;
     let a be Element of R, b be Element of S;
     assume
A1:  a = b;
A2:  R is Subring of S by Def1;
A3:  the carrier of Polynom-Ring R c= the carrier of Polynom-Ring S by Th6;
     set q = rpoly(1,b);
     reconsider p = rpoly(1,a) as Element of the carrier
     of Polynom-Ring R by POLYNOM3:def 10;
     reconsider p as Element of the carrier of Polynom-Ring S by A3;
     reconsider p as Polynomial of S;
     len q > 0 by UPROOTS:17; then
A4:  len q -' 1 = len q - 1 by XREAL_0:def 2; then
     reconsider lenq = len q - 1 as Element of NAT;
A5:  1 = deg q by HURWITZ:27 .= len q - 1 by HURWITZ:def 2; then
A6:  len q = 1 + 1;
A7:   now let i be Nat;
       assume
A8:    i >= len q;
       reconsider j = i as Element of NAT by ORDINAL1:def 12;
       j <> 0 & j <> 1 by A8,A6; then
       rpoly(1,a).j = 0.R by HURWITZ:26;
       hence p.i = 0.S by A2,C0SP1:def 3;
     end;
     now let m be Nat;
       assume
A9:    m is_at_least_length_of p;
       reconsider j = lenq as Element of NAT;
       now assume len q > m; then
         lenq + 1 > m; then
         lenq >= m by NAT_1:13; then
A10:     p.(len q-'1) = 0.S by A4,A9,ALGSEQ_1:def 2;
         rpoly(1,a).1 = 1_R by HURWITZ:25 .= 1.S by A2,C0SP1:def 3;
         hence contradiction by A10,A5,XREAL_0:def 2;
       end;
       hence len q <= m;
     end; then
A11: len p = len q by A7,ALGSEQ_1:def 3,ALGSEQ_1:def 2;
     now let k be Nat;
       assume
A12:   k < len q;
       len q - 1 = deg q by HURWITZ:def 2 .= 1 by HURWITZ:27; then
       k < 1 + 1 by A12; then
A13:   k <= 1 by NAT_1:13;
       per cases;
         suppose
A14:       k = 0; then
A15:       rpoly(1,a).k = -power(R).(a,0+1) by HURWITZ:25
           .= -(power(R).(a,0) * a) by GROUP_1:def 7
           .= -(1_R * a) by GROUP_1:def 7
           .= -b by A1,Lm2;
           q.k = -power(S).(b,0+1) by A14,HURWITZ:25
           .= -(power(S).(b,0) * b) by GROUP_1:def 7
           .= -(1_S * b) by GROUP_1:def 7
           .= -b;
           hence p.k = q.k by A15;
         end;
         suppose k <> 0; then
           k + 1 > 0 + 1 by XREAL_1:6; then
A16:       k >= 1 by NAT_1:13; then
A17:       k = 1 by A13,XXREAL_0:1;
           rpoly(1,a).1 = 1_R by HURWITZ:25 .= 1.S by A2,C0SP1:def 3;
           hence p.k = 1_S by A16,A13,XXREAL_0:1 .= q.k by
           A17,HURWITZ:25;
         end;
       end;
       hence thesis by A11,ALGSEQ_1:12;
     end;
