
theorem
for F being Field
for K being FieldExtension of F
for E being K-extending FieldExtension of F
st E is F-separable holds E is K-separable & K is F-separable
proof
let F be Field, K be FieldExtension of F;
let E be K-extending FieldExtension of F;
assume AS: E is F-separable; then
A: E is K-algebraic & K is F-algebraic by FIELD_7:40;
now let a be Element of E;
  reconsider b = a as K-algebraic Element of E by AS;
  reconsider c = a as F-algebraic Element of E by AS;
  set f = MinPoly(c,F), g = MinPoly(b,K);
  f is Polynomial of K by FIELD_4:8; then
  reconsider f1 = f as Element of the carrier of Polynom-Ring K
                                                        by POLYNOM3:def 10;
  deg f > 0 by RING_4:def 4; then
  deg f1 > 0 by FIELD_4:20; then
  reconsider f1 = f as non constant Element of the carrier of Polynom-Ring K
    by RING_4:def 4;
  K: f1 is separable by AS,YY;
  0.E = Ext_eval(f,c) by FIELD_6:51 .= Ext_eval(f1,b) by FIELD_7:15; then
  consider r being Polynomial of K such that
  H: g *' r = f1 by RING_4:1,FIELD_6:53;
  g is non constant Element of the carrier of Polynom-Ring K &
  r is Element of the carrier of Polynom-Ring K by POLYNOM3:def 10; then
  g is separable by H,K,ThSepPR;
  hence a is K-separable;
  end;
hence E is K-separable by AS,FIELD_7:40;
now let a be Element of K;
  reconsider b = a as F-algebraic Element of K by A;
  K is Subfield of E by FIELD_4:7; then
  the carrier of K c= the carrier of E by EC_PF_1:def 1; then
  reconsider c = b as F-algebraic Element of E by AS;
  MinPoly(b,F) = MinPoly(c,F) by FIELD_10:10;
  hence a is F-separable by AS;
  end;
hence K is F-separable by AS,FIELD_7:40;
end;
