
theorem inS:
for F being Field, E being FieldExtension of F,
    L being F-monomorphic Field,
    f being Monomorphism of F,L
for K being strict FieldExtension of F,
    g being Function of K,L st g is monomorphism
holds [K,g] in Ext_Set(f,E) iff
      (E is FieldExtension of K & F is Subfield of K & g is f-extending)
proof
let F be Field, E be FieldExtension of F,
    L be F-monomorphic Field, f be Monomorphism of F,L;
let K be strict FieldExtension of F, g be Function of K,L;
assume AS: g is monomorphism;
A: now assume [K,g] in Ext_Set(f,E); then
     consider K2 be Element of SubFields(E), g2 be Function of K2,L such that
     A1: [K,g] = [K2,g2] &
         ex K1 being FieldExtension of F, g1 being Function of K1,L
         st K1 = K2 & g1 = g2 & g1 is monomorphism f-extending;
     consider K1 being FieldExtension of F, g1 being Function of K1,L such that
     A2: K1 = K2 & g1 = g2 & g1 is monomorphism f-extending by A1;
     A3: K2 = [K,g]`1 by A1,XTUPLE_0:def 2 .= K;
     A7: g2 = [K,g]`2 by A1,XTUPLE_0:def 3 .= g;
     K is strict Subfield of E by A3,subfie;
     hence E is FieldExtension of K & F is Subfield of K &
           g is f-extending by A2,A7,FIELD_4:7;
     end;
now assume B0:
  E is FieldExtension of K & F is Subfield of K & g is f-extending; then
  K is Subfield of E by FIELD_4:7;
  then K in SubFields(E) by subfie;
  hence [K,g] in Ext_Set(f,E) by B0,AS;
  end;
hence thesis by A;
end;
