
theorem lemNor1d:
for F being Field,
    E1,E2 being FieldExtension of F
for p being non constant Element of the carrier of Polynom-Ring F
st E1 == E2 holds E1 is SplittingField of p implies E2 is SplittingField of p
proof
let F be Field, E1,E2 be FieldExtension of F;
let p be non constant Element of the carrier of Polynom-Ring F;
assume AS: E1 == E2;
assume B: E1 is SplittingField of p; then
C: p splits_in E1 by FIELD_8:def 1;
now let E be FieldExtension of F;
  assume D: p splits_in E & E is Subfield of E2; then
     E2 is FieldExtension of E by FIELD_4:7; then
     E1 is FieldExtension of E by AS,lemNor1cu; then
     E is Subfield of E1 by FIELD_4:7; then
     E == E1 by B,D,FIELD_8:def 1;
  hence E == E2 by AS,helpb;
  end;
hence thesis by AS,lemNor1c,C,FIELD_8:def 1;
end;
