
theorem thiso:
for F being Field,
    Z being set
for r being Renaming of (the carrier of F), Z
holds r" is additive multiplicative unity-preserving
            one-to-one onto
proof
let F be Field, Z be set;
let r be Renaming of (the carrier of F), Z;
set K = RenField r;
H0: dom r = the carrier of F by defr;
now let a,b be Element of K;
  reconsider a1 = a, b1 = b as Element of (rng r) by defrf;
  thus (r").(a+b) = (r").((ren_add r).(a1,b1)) by defrf
        .= (r").(r.((r").a + (r").b)) by defra
        .= (r").a + (r").b by H0,FUNCT_1:34;
  end;
hence r" is additive;
now let a,b be Element of K;
  reconsider a1 = a, b1 = b as Element of (rng r) by defrf;
  thus (r").(a*b) = (r").((ren_mult r).(a1,b1)) by defrf
        .= (r").(r.((r").a * (r").b)) by defrm
        .= (r").a * (r").b by H0,FUNCT_1:34;
  end;
hence r" is multiplicative;
r.(1.F) = 1.K by defrf;
hence r" is unity-preserving by H0,FUNCT_1:34;
thus r" is one-to-one;
thus r" is onto by lemonto;
end;
