reserve i1 for Element of INT;
reserve j1,j2,j3 for Integer;
reserve p,s,k,n for Nat;
reserve x,y,xp,yp for set;
reserve G for Group;
reserve a,b for Element of G;
reserve F for FinSequence of G;
reserve I for FinSequence of INT;

theorem Th14:
  for n be non zero Nat holds 1_INT.Group(n) = 0
proof
  let n be non zero Nat;
  reconsider e = 0 as Element of Segm(n) by NAT_1:44;
  reconsider e as Element of INT.Group(n);
  for h being Element of INT.Group(n) holds h * e = h & e * h = h
  proof
    let h be Element of INT.Group(n);
    reconsider A = h as Element of Segm(n);
    reconsider A as Element of NAT;
A1: A < n by NAT_1:44;
A2: e * h = (0 + A) mod n by Def4
      .= h by A1,NAT_D:24;
    h * e = (A + 0) mod n by Def4
      .= h by A1,NAT_D:24;
    hence thesis by A2;
  end;
  hence thesis by GROUP_1:4;
end;
