
theorem helpp:
for R being comRing,
    S being comRingExtension of R
for x1,x2 being Element of S, y1,y2 being Element of R
for n being Element of NAT
st x1 = y1 & x2 = y2 holds <%x1,x2%>`^n = <%y1,y2%>`^n
proof
let R be comRing, S be comRingExtension of R;
let x1,x2 be Element of S, y1,y2 be Element of R;
let n be Element of NAT;
assume A: x1 = y1 & x2 = y2;
defpred P[Nat] means <%x1,x2%>`^($1) = <%y1,y2%>`^($1);
<%x1,x2%>`^0 = 1_.(S) by POLYNOM5:15
            .= 1_.(R) by FIELD_4:14
            .= <%y1,y2%>`^0 by POLYNOM5:15; then
IA: P[0];
reconsider qS = <%x1,x2%> as Element of the carrier of Polynom-Ring S
  by POLYNOM3:def 10;
reconsider qR = <%y1,y2%> as Element of the carrier of Polynom-Ring R
  by POLYNOM3:def 10;
R is Subring of S by FIELD_4:def 1; then
E: <%x1,x2%> = <%y1,y2%> by A,pr20;
IS: now let k be Nat;
    assume IV: P[k];
    B: <%x1,x2%>`^(k+1)
          = (power Polynom-Ring S).(<%x1,x2%>,k+1) by POLYNOM5:def 3
         .= (power Polynom-Ring S).(qS,k) * qS by GROUP_1:def 7;
    C: <%y1,y2%>`^(k+1)
          = (power Polynom-Ring R).(<%y1,y2%>,k+1) by POLYNOM5:def 3
         .= (power Polynom-Ring R).(qR,k) * qR by GROUP_1:def 7;
    D: (power Polynom-Ring R).(qR,k)
          = <%x1,x2%>`^k by IV,POLYNOM5:def 3
         .= (power Polynom-Ring S).(qS,k) by POLYNOM5:def 3;
    reconsider u = (power Polynom-Ring R).(qR,k) as Polynomial of R
       by POLYNOM3:def 10;
    reconsider v = (power Polynom-Ring S).(qS,k) as Polynomial of S
       by POLYNOM3:def 10;
    (power Polynom-Ring R).(qR,k) * qR
         = u *' <%y1,y2%> by POLYNOM3:def 10
        .= v *' <%x1,x2%> by D,E,FIELD_4:17
        .= (power Polynom-Ring S).(qS,k) * qS by POLYNOM3:def 10;
    hence P[k+1] by B,C;
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
