
theorem splift:
for F1 being Field,
    p1 being non constant Element of the carrier of Polynom-Ring F1
for F2 being FieldExtension of F1,
    p2 being non constant Element of the carrier of Polynom-Ring F2
for E being SplittingField of p1
st p2 = p1 & E is F2-extending holds E is SplittingField of p2 
proof
let F1 be Field,
    p1 be non constant Element of the carrier of Polynom-Ring F1;
let F2 be FieldExtension of F1,
    p2 be non constant Element of the carrier of Polynom-Ring F2;
let E be SplittingField of p1;
assume AS: p2 = p1 & E is F2-extending;
p1 splits_in E by defspl; then 
consider a being non zero Element of E, q being Ppoly of E such that
A: p1 = a * q by FIELD_4:def 5;
now let K be FieldExtension of F2;
  assume C: p2 splits_in K & K is Subfield of E; then 
  consider a being non zero Element of K, q being Ppoly of K such that
  D: p2 = a * q by FIELD_4:def 5;
  p1 splits_in K by D,AS,FIELD_4:def 5;
  hence K == E by C,defspl;
  end;
hence thesis by A,AS,FIELD_4:def 5,defspl;
end;
