reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem Th29:
  0 <= r & r <= 2*PI & cos r = -1 implies r = PI
proof
  assume that
A1: 0 <= r and
A2: r <= 2*PI and
A3: cos r = -1;
A4: r = 2*PI or r < 2*PI by A2,XXREAL_0:1;
  thus r = PI + 2*PI*[\r/(2*PI)/] by A3,Th25
    .= PI + 2*PI*0 by A1,A3,A4,Th1,SIN_COS:77
    .= PI;
end;
