
theorem 
for F being Field,
    E1,E2 being FieldExtension of F
st E1,E2 are_isomorphic_over F holds E2,E1 are_isomorphic_over F
proof
let F be Field, E1,E2 be FieldExtension of F;
F is Subring of E1 by FIELD_4:def 1; then
H1: the carrier of F c= the carrier of E1 by C0SP1:def 3;
assume E1,E2 are_isomorphic_over F; then
consider f being Function of E1,E2 such that A: f is F-isomorphism;
B: f is F-fixing isomorphism by A; 
C: f is one-to-one & f is onto & rng f = the carrier of E2 by A,FUNCT_2:def 3;
then reconsider g = f" as Function of E2,E1 by FUNCT_2:25;
now let a be Element of F;
  H2: dom f = the carrier of E1 by FUNCT_2:def 1;
  H3: a in the carrier of E1 by H1;
  thus g.a = g.(f.a) by B .= a by H2,H3,A,FUNCT_1:34;
  end;
then D: g is F-fixing;
H4: f is additive multiplicative unity-preserving by A;
H5: g is additive
    proof
    now let a,b be Element of E2;
    consider x being object such that
    A2: x in the carrier of E1 & a = f.x by C,FUNCT_2:11;
    reconsider x as Element of E1 by A2;
    consider y being object such that
    A3: y in the carrier of E1 & b = f.y by C,FUNCT_2:11;
    reconsider y as Element of E1 by A3;
    thus g.a + g.b = x + g.(f.y) by A2,A3,A,FUNCT_2:26
             .= x + y by A,FUNCT_2:26
             .= g.(f.(x+y)) by A,FUNCT_2:26
             .= g.(a+b) by H4,A2,A3;
    end;
    hence thesis;
    end;
H6: g is multiplicative
    proof
    now let a,b be Element of E2;
    consider x being object such that
    A2: x in the carrier of E1 & a = f.x by C,FUNCT_2:11;
    reconsider x as Element of E1 by A2;
    consider y being object such that
    A3: y in the carrier of E1 & b = f.y by C,FUNCT_2:11;
    reconsider y as Element of E1 by A3;
    thus g.a * g.b = x * g.(f.y) by A,A2,A3,FUNCT_2:26
             .= x * y by A,FUNCT_2:26
             .= g.(f.(x*y)) by A,FUNCT_2:26
             .= g.(a*b) by H4,A2,A3;
    end;
    hence thesis;
    end;
F: g is unity-preserving by H4,FUNCT_2:26;
now let x be object;
  H7: now assume H8: x in rng g;
      rng g c= the carrier of E1 by RELAT_1:def 19;
      hence x in the carrier of E1 by H8;
      end;
  now assume x in the carrier of E1;
     then reconsider x1 = x as Element of E1;
     f.x1 in the carrier of E2;
     then A9: f.x1 in dom g by FUNCT_2:def 1;
     g.(f.x1) = x1 by A,FUNCT_2:26;
     hence x in rng g by A9,FUNCT_1:def 3;
     end;
  hence x in rng g iff x in the carrier of E1 by H7;
  end;
then g is onto by TARSKI:2;
then g is F-isomorphism by D,F,H5,H6;
hence E2,E1 are_isomorphic_over F;
end;
