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 Th4:
  0 < r < 2 * PI implies sin (r/2) <> 0
  proof
    assume
A1: 0 < r < 2 * PI;
    assume
A2: sin(r/2) = 0;
    consider i0 be Integer such that
A3: r / 2 = PI * i0 by A2,BORSUK_7:7;
A4: r = 2 * i0 * PI by A3;
    reconsider p = 2*PI as Real;
    0 < i0 * p < p by A1,A4;
    then i0 = 1 by Th1;
    hence thesis by A1,A3;
  end;
