
theorem m4spl:
for R being Ring,
    S being RingExtension of R
for p being Element of the carrier of Polynom-Ring R
for q being Element of the carrier of Polynom-Ring S st p = q
for T1 being RingExtension of S,
    T2 being RingExtension of R
st T1 = T2 holds Roots(T2,p) = Roots(T1,q)
proof
let R be Ring, S be RingExtension of R;
let p be Element of the carrier of Polynom-Ring R;
let q be Element of the carrier of Polynom-Ring S;
assume AS1: p = q;
let T1 be RingExtension of S, T2 be RingExtension of R;
assume AS2: T1 = T2;
the carrier of Polynom-Ring S c= the carrier of Polynom-Ring T1
   by FIELD_4:10; then
reconsider q1 = q as Element of the carrier of Polynom-Ring T1;
the carrier of Polynom-Ring R c= the carrier of Polynom-Ring T2
   by FIELD_4:10; then
reconsider p1 = p as Element of the carrier of Polynom-Ring T2;
thus Roots(T2,p) = Roots(p1) by FIELD_7:13
                .= Roots(T1,q) by AS1,AS2,FIELD_7:13;
end;
