reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem Th2:
  sin(x + (2*n+1)*PI) = -sin x
proof
  defpred X[Nat] means sin(x + (2*$1+1)* PI) = -sin x;
A1: for n being Nat st X[n] holds X[n+1]
  proof
    let n be Nat;
    assume
A2: sin(x + (2*n+1)*PI) = -sin x;
    sin(x + (2*(n+1)+1)*PI)=sin((x + (2*n+1)*PI)+2*PI)
      .=sin(x + (2*n+1)*PI)*cos(2*PI) + cos(x + (2*n+1)*PI)*sin(2*PI) by
SIN_COS:75
      .= -sin(x) by A2,SIN_COS:77;
    hence thesis;
  end;
A3: X[0] by SIN_COS:79;
  for n being Nat holds X[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
