reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem
  PI/2 + 2*PI*i <= r <= PI + 2*PI*i & r/PI is rational & cos r is rational
  implies
  r in { PI/2+2*PI*i , 2*PI/3+2*PI*i , PI+2*PI*i }
  proof
    set a = 2*PI*i;
    set R = r-a;
    assume PI/2+a <= r <= PI+a; then
A1: PI/2+a-a <= R <= PI+a-a by XREAL_1:9;
    assume
A2: r/PI is rational & cos r is rational;
    a/PI = (2*i*PI)/PI
    .= 2*i by XCMPLX_1:89; then
A3: R/PI = r/PI-2*i;
    R = 2*PI*(-i)+r;
    then cos r = cos R by COMPLEX2:9;
    then R in {PI/2,2*PI/3,PI} by A1,A2,A3,Th55;
    then R = PI/2 or R = 2*PI/3 or R = PI by ENUMSET1:def 1;
    hence thesis by ENUMSET1:def 1;
  end;
