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 Th59:
  3*PI/2 <= r <= 2*PI & r/PI is rational & cos r is rational implies
  r in {3*PI/2,5*PI/3,2*PI}
  proof
    set R = 2*PI-r;
    assume 3*PI/2 <= r <= 2*PI;
    then
A1: 2*PI-2*PI <= R <= 2*PI-3*PI/2 by XREAL_1:13;
    assume
A2: r/PI is rational & cos r is rational;
A3: R/PI = 2*PI/PI-r/PI
    .= 2-r/PI by XCMPLX_1:89;
    cos R = cos r by EUCLID10:4;
    then R in {0,PI/3,PI/2} by A1,A2,A3,Th53;
    then R = 0 or R = PI/3 or R = PI/2 by ENUMSET1:def 1;
    hence thesis by ENUMSET1:def 1;
  end;
