
theorem ThSep02:
for F being Field,
    p being non constant Element of the carrier of Polynom-Ring F
holds p is separable iff
      for E being FieldExtension of F
      for a being Element of E holds multiplicity(p,a) <= 1
proof
let F be Field, p be non constant Element of the carrier of Polynom-Ring F;
A: now assume AS: p is separable;
   now let E be FieldExtension of F, a be Element of E;
     assume B: multiplicity(p,a) > 1;
     the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
        by FIELD_4:10; then
     reconsider p1 = p as Element of the carrier of Polynom-Ring E;
     deg p > 0 by RING_4:def 4; then
     deg p1 > 0 by FIELD_4:20; then
     reconsider p1 as non constant Element of the carrier of Polynom-Ring E
        by RING_4:def 4;
     consider K being FieldExtension of E such that
     D: p1 splits_in K by FIELD_5:31;
     reconsider K as E-extending FieldExtension of F;
     multiplicity(p1,a) > 1 by B,FIELD_14:def 6; then
     multiplicity(p1,@(a,K)) > 1 by multi3; then
     E: multiplicity(p,@(a,K)) > 1 by sepsep;
     H: @(a,K) = a by FIELD_7:def 4;
     consider b being non zero Element of K, q being Ppoly of K such that
     I: p1 = b * q by D,FIELD_4:def 5;
     F: p splits_in K by I,FIELD_4:def 5;
     E is Subfield of K by FIELD_4:7; then
     0.K = 0.E by EC_PF_1:def 1
        .= Ext_eval(p,a) by B,mulzero,FIELD_4:def 2
        .= Ext_eval(p,@(a,K)) by H,FIELD_6:11;
     hence contradiction by F,E,AS,ThSep0,FIELD_4:def 2;
     end;
   hence for E being FieldExtension of F
         for a being Element of E holds multiplicity(p,a) <= 1;
   end;
now assume AS: for E being FieldExtension of F
               for a being Element of E holds multiplicity(p,a) <= 1;
  consider E being FieldExtension of F such that
  B: p splits_in E by FIELD_5:31;
  now let a be Element of E;
    assume a is_a_root_of p,E; then
    multiplicity(p,a) >= 1 & multiplicity(p,a) <= 1 by AS,mulzero;
    hence multiplicity(p,a) = 1 by XXREAL_0:1;
    end;
  hence p is separable by B,ThSep1;
  end;
hence thesis by A;
end;
