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 Th57:
  PI <= r <= 3*PI/2 & r/PI is rational & cos r is rational implies
  r in {PI,4*PI/3,3*PI/2}
  proof
    set R = r-PI;
    assume PI <= r <= 3*PI/2;
    then
A1: PI-PI <= R <= 3*PI/2-PI by XREAL_1:13;
    assume
A2: r/PI is rational & cos r is rational;
A3: R/PI = r/PI-PI/PI
    .= r/PI-1 by XCMPLX_1:60;
    cos R = cos(-(PI-r))
    .= cos(PI-r) by SIN_COS:31
    .= -cos r by EUCLID10:2;
    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;
