reserve n for Nat;
reserve i for Integer;
reserve r,s,t for Real;
reserve An,Bn,Cn,Dn for Point of TOP-REAL n;
reserve L1,L2 for Element of line_of_REAL n;
reserve A,B,C for Point of TOP-REAL 2;

theorem Th5:
  -2 * PI < r < 0 implies sin (r/2) <> 0
  proof
    assume
A1: -2 * PI < r < 0;
    assume
A2: sin(r/2) = 0;
    0 * (-1) < r * (-1) & r * (-1) < (-2 * PI) * (-1) by A1,XREAL_1:69;
    then
A3: sin ((-r)/2) <> 0 by Th4;
    reconsider r0 = r/2 as Real;
    sin (-r0) = - sin (r0) by SIN_COS:31;
    hence contradiction by A2,A3;
  end;
