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
  r/PI is rational & cos r is rational implies cos r in {0,1,-1,1/2,-1/2}
  proof
    consider i such that
A0: 2*PI*i <= r <= 2*PI*(i+1) by Th16;
    set a = 2*PI*i;
    set R = r-a;
A2: a-a <= R <= 2*PI+a-a by A0,XREAL_1:9;
    assume
A3: r/PI is rational & cos r is rational;
    a/PI = (2*i*PI)/PI
    .= 2*i by XCMPLX_1:89;
    then
A4: R/PI = r/PI-2*i;
    R = 2*PI*(-i)+r;
    then cos r = cos R by COMPLEX2:9;
    hence thesis by A2,A3,A4,Lm4;
  end;
