reserve p,q,r,th,th1 for Real;
reserve n for Nat;

theorem
  for n being Nat holds sin.th = sin.(2*PI*n + th)
proof
  defpred X[Nat] means for th holds sin.th = sin.(2*PI*$1 + th);
  let n be Nat;
A1: for n being Nat st X[n] holds X[n+1]
  proof
    let n be Nat such that
A2: for th holds sin.th = sin.(2*PI*n + th);
    for th holds sin.th = sin.(2*PI*(n+1) + th)
    proof
      let th;
      sin.(2*PI*(n+1) + th) = sin.((2*PI*n+th) + 2*PI)
        .= sin.(2*PI*n + th) by SIN_COS:78
        .= sin.th by A2;
      hence thesis;
    end;
    hence thesis;
  end;
A3: X[0];
  for n being Nat holds X[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
