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*PI*i <= r <= 3*PI/2 + 2*PI*i & r/PI is rational & sin r is rational
  implies
  r in { PI+2*PI*i , 7*PI/6+2*PI*i , 3*PI/2+2*PI*i }
  proof
    set a = 2*PI*i;
    set R = r-a;
    assume PI+a <= r <= 3*PI/2+a;
    then
A1: PI+a-a <= R <= 3*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 {PI,7*PI/6,3*PI/2} by A1,A2,A3,Th66;
    then R = PI or R = 7*PI/6 or R = 3*PI/2 by ENUMSET1:def 1;
    hence thesis by ENUMSET1:def 1;
  end;
