
theorem lemma2e:
for R being non degenerated Ring,
    S being RingExtension of R
for p being Polynomial of R
for q being Polynomial of S st p = q holds LC p = LC q
proof
let R be non degenerated Ring, S be RingExtension of R;
let p be Polynomial of R; let q be Polynomial of S;
assume A: p = q;
H: R is Subring of S by FIELD_4:def 1;
per cases;
suppose C: p = 0_.(R);
D: LC p = p.(len p -'1) by RATFUNC1:def 6
       .= 0.S by C,H,C0SP1:def 3;
E: q = 0_.(S) by A,C,FIELD_4:12;
thus LC q = q.(len q -'1) by RATFUNC1:def 6 .= LC p by D,E;
end;
suppose F: p <> 0_.(R); then
C: p is non zero by UPROOTS:def 5;
D: now assume q is zero; then
   q = 0_.(S) by UPROOTS:def 5 .= 0_.(R) by FIELD_4:12;
   hence contradiction by A,F;
   end;
E: p is Element of the carrier of Polynom-Ring R &
   q is Element of the carrier of Polynom-Ring S by POLYNOM3:def 10;
thus LC p = p.(deg p) by C,FIELD_6:2
         .= q.(deg q) by A,E,FIELD_4:20
         .= LC q by D,FIELD_6:2;
end;
end;
