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
  2*PI*i <= r <= PI/2 + 2*PI*i & r/PI is rational & sin r is rational implies
  r in { 2*PI*i , PI/6+2*PI*i , PI/2+2*PI*i }
  proof
    set a = 2*PI*i;
    set R = r-a;
    assume a <= r <= PI/2+a;
    then
A1: a-a <= R <= PI/2+a-a by XREAL_1:9;
    assume
A2: r/PI is rational & sin 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 sin r = sin R by COMPLEX2:8;
    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;
