
theorem lemNor1c:
for F being Field,
    E1,E2 being FieldExtension of F
for p being Polynomial of F
st E1 == E2 holds p splits_in E1 implies p splits_in E2
proof
let F be Field, E1,E2 be FieldExtension of F, p be Polynomial of F;
assume AS: E1 == E2; then
A: the doubleLoopStr of E1 = the doubleLoopStr of E2 by FIELD_7:def 1;
assume p splits_in E1; then
consider a being non zero Element of E1, q being Ppoly of E1 such that
C: p = a * q by FIELD_4:def 5;
reconsider q1 = q as Ppoly of E2 by AS,lemNor1cx;
reconsider a1 = a as Element of E2 by A;
now assume a1 = 0.E2;
  then a1 = 0.E1 by A;
  hence contradiction;
  end; then
reconsider a1 = a as non zero Element of E2 by STRUCT_0:def 12;
p = a1 * q1 by AS,C,lemNor1cy;
hence p splits_in E2 by FIELD_4:def 5;
end;
