
theorem
for F being Field
for A1,A2 being AlgebraicClosure of F holds A1,A2 are_isomorphic_over F
proof
let F be Field, E1,E2 be AlgebraicClosure of F;
reconsider L = E2 as F-monomorphic F-homomorphic algebraic-closed Field;
reconsider f = id F as Monomorphism of F,L by lift5a;
consider g being Function of E1,L such that
A: g is monomorphism f-extending by lift2;
C: the doubleLoopStr of F == F by lemug1;
Image f = the doubleLoopStr of F
   proof
   set If = Image f;
   H: F is Subfield of E2 by FIELD_4:7;
   B: the carrier of If = rng f by RING_2:def 6 .= the carrier of F;
   C: the addF of If = (the addF of L)||(rng f) by RING_2:def 6
                    .= the addF of F by H,EC_PF_1:def 1;
   D: the multF of If = (the multF of L)||(rng f) by RING_2:def 6
                     .= the multF of F by H,EC_PF_1:def 1;
   E: the OneF of If = 1.L by RING_2:def 6 .= 1.F by H,EC_PF_1:def 1;
   the ZeroF of If = 0.L by RING_2:def 6 .= 0.F by H,EC_PF_1:def 1;
   hence thesis by B,C,D,E;
   end; then
L is AlgebraicClosure of (Image f) by C,lift4;
then B: g is isomorphism by A,lift3;
now let a be Element of F;
  thus g.a = f.a by A .= a;
  end; then
g is F-fixing;
hence thesis by B,FIELD_8:def 5;
end;
