
theorem eval2:
for R being non degenerated Ring,
    S being RingExtension of R
for a1,b1,c1 being Element of R,
    a2,b2,c2 being Element of S st a1 = a2 & b1 = b2 & c1 = c2
holds <%c2,b2,a2%> = <%c1,b1,a1%>
proof
let R be non degenerated Ring, S be RingExtension of R;
let a1,b1,c1 be Element of R, a2,b2,c2 be Element of S;
assume A: a1 = a2 & b1 = b2 & c1 = c2;
set p = <%c2,b2,a2%>, q = <%c1,b1,a1%>;
H: R is Subring of S by FIELD_4:def 1;
B: len p <= 3 & len q <= 3 by qua2;
   now let i be Element of NAT;
    i <= 2 implies i = 0 or ... or i = 2; then
    per cases;
    suppose C: i = 0;
      hence p.i = c1 by A,qua1 .= q.i by C,qua1;
      end;
    suppose C: i = 1;
      hence p.i = b1 by A,qua1 .= q.i by C,qua1;
      end;
    suppose C: i = 2;
      hence p.i = a1 by A,qua1 .= q.i by C,qua1;
      end;
    suppose i > 2;
      then i + 1 > 2 + 1 by XREAL_1:6;
      then C: i >= 3 by NAT_1:13;
      hence p.i = 0.S by B,XXREAL_0:2,ALGSEQ_1:8
               .= 0.R by H,C0SP1:def 3
               .= q.i by C,B,XXREAL_0:2,ALGSEQ_1:8;
      end;
    end;
hence thesis;
end;
