theorem Th13:
  for i,j,m being Integer st i mod m = j mod m holds i|^n mod m = j|^n mod m
proof
  let i,j,m be Integer;
  defpred P[Nat] means i|^$1 mod m = j|^$1 mod m;
  assume
A1: i mod m = j mod m;
A2: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume
A3: P[n];
    thus i|^(n+1) mod m = ((i|^n)*i) mod m by NEWTON:6
      .= (((j|^n) mod m)*(j mod m)) mod m by A1,A3,NAT_D:67
      .= ((j|^n)*j) mod m by NAT_D:67
      .= j|^(n+1) mod m by NEWTON:6;
  end;
  i|^0 = 1 by NEWTON:4;
  then
A4: P[0] by NEWTON:4;
   for n being Nat holds P[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
