reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve p for Prime;
reserve a,b,c,d for Element of GF(p);
reserve K for Ring;
reserve a1,a2,a3,a4,a5,a6 for Element of K;

theorem Th5:
  n >= 1 implies (0.K) |^ n = 0.K
  proof
    set a1 = 0.K;
    assume A2: n >= 1;
    n - 1 in NAT by A2,INT_1:5;
    then consider n1 be Nat such that
A3: n1 = n - 1;
    a1 |^n = a1 |^(n1+1) by A3
    .= a1|^n1 * a1 by EC_PF_1:24;
    hence a1 |^n = 0.K;
  end;
