
theorem MG:
for R being Skew-Field
for n being Nat
for a being Element of R
for b being Element of MultGroup R st a = b holds a|^n = b|^n
proof
let R be Skew-Field, n be Nat, a be Element of R,
    b be Element of MultGroup R;
set M = MultGroup R;
assume AS: a = b;
defpred P[Nat] means
  for a being Element of R, b being Element of M
  st a = b holds a|^($1) = b|^($1);
IA: P[0]
    proof
    now let a be Element of R, b be Element of M;
      thus a|^0 = 1_R by BINOM:8 .= 1_M by UNIROOTS:17 .= b|^0 by GROUP_1:25;
      end;
    hence thesis;
    end;
IS: now let k be Nat;
    assume A: P[k];
    now let a be Element of R, b be Element of M;
      assume B: a = b; then
      C: a|^k = b|^k by A;
      thus a|^(k+1)
             = a|^k * a|^1 by BINOM:10
            .= a|^k * a by BINOM:8
            .= b|^k * b by B,C,UNIROOTS:16
            .= b|^(k+1) by GROUP_1:34;
      end;
    hence P[k+1];
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis by AS;
end;
