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 Th64:
  PI/2 <= r <= PI & r/PI is rational & sin r is rational implies
  r in {PI/2,5*PI/6,PI}
  proof
    set R = PI-r;
    assume PI/2 <= r <= PI;
    then
A1: PI-PI <= R <= PI-PI/2 by XREAL_1:13;
    assume
A2: r/PI is rational & sin r is rational;
A3: R/PI = PI/PI-r/PI
    .= 1-r/PI by XCMPLX_1:60;
    sin R = sin r by EUCLID10:1;
    then R in {0,PI/6,PI/2} by A1,A2,A3,Th62;
    then R = 0 or R = PI/6 or R = PI/2 by ENUMSET1:def 1;
    hence thesis by ENUMSET1:def 1;
  end;
