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 Th62:
  0 <= r <= PI/2 & r/PI is rational & sin r is rational implies
  r in {0,PI/6,PI/2}
  proof
    set t = PI/2-r;
    assume 0 <= r;
    then
A1: t <= PI/2-0 by XREAL_1:10;
    assume r <= PI/2;
    then
A2: PI/2-PI/2 <= t by XREAL_1:10;
    assume
A3: r/PI is rational & sin r is rational;
A4: t/PI = PI/2/PI-r/PI;
A5: PI/2/PI = 1/2 by XCMPLX_1:203;
    cos t = cos(PI/2)*cos(r)+sin(PI/2)*sin(r) by SIN_COS:83
    .= sin r by SIN_COS:77;
    then t in {0,PI/3,PI/2} by A1,A2,A3,A4,A5,Th53;
    then t = 0 or t = PI/3 or t = PI/2 by ENUMSET1:def 1;
    hence thesis by ENUMSET1:def 1;
  end;
