
theorem lemquadr:
for R being domRing,
    S being domRingExtension of R
for a being Element of R,
    b being Element of S st b^2 = a^2 holds b = a or b = -a
proof
let R be domRing,S be domRingExtension of R;
let a be Element of R, b be Element of S;
assume AS: b^2 = a^2;
A: R is Subring of S by FIELD_4:def 1; then
the carrier of R c= the carrier of S by C0SP1:def 3; then
reconsider a1 = a as Element of S;
a1^2 = a1 * a1 by O_RING_1:def 1
    .= a * a by A,FIELD_6:16
    .= b^2 by AS,O_RING_1:def 1; then
b = a1 or b = -a1 by REALALG2:10;
hence thesis by A,FIELD_6:17;
end;
