
theorem lift3:
for F being Field,
    L being F-monomorphic F-homomorphic Field,
    A being AlgebraicClosure of F
for f being Monomorphism of F,L st L is AlgebraicClosure of (Image f)
for g being Function of A,L
st g is monomorphism f-extending holds g is isomorphism
proof
let F be Field, L be F-monomorphic F-homomorphic Field,
    E being AlgebraicClosure of F, f be Monomorphism of F,L;
assume AS1: L is AlgebraicClosure of (Image f);
let g be Function of E,L;
assume AS2: g is monomorphism f-extending; then
reconsider L1 = L as E-monomorphic E-homomorphic Field
                                     by RING_3:def 3,RING_2:def 4;
reconsider g1 = g as Monomorphism of E,L1 by AS2;
B: Image g1 is algebraic-closed by lift3c;
reconsider If = Image f as Field;
If is Subfield of Image g1 by AS2,lift3b; then
reconsider Ig = Image g1 as FieldExtension of If by FIELD_4:7;
reconsider L1 as FieldExtension of If by AS1;
Ig is Subring of L1 by FIELD_5:12; then
reconsider L1 as Ig-extending FieldExtension of If by FIELD_4:def 1;
L1 is If-algebraic by AS1; then
Ig is AlgebraicClosure of If by B,defAC,FIELD_7:40; then
the doubleLoopStr of L1 = the doubleLoopStr of Ig
     by AS1,lift3a,FIELD_7:def 1; then
g is onto by RING_2:def 6;
hence thesis by AS2;
end;
