let r be Real; :: thesis: for i being Integer st (PI / 2) + ((2 * PI) * i) <= r & r <= PI + ((2 * PI) * i) & r / PI is rational & cos r is rational holds
r in {((PI / 2) + ((2 * PI) * i)),(((2 * PI) / 3) + ((2 * PI) * i)),(PI + ((2 * PI) * i))}
let i be Integer; :: thesis: ( (PI / 2) + ((2 * PI) * i) <= r & r <= PI + ((2 * PI) * i) & r / PI is rational & cos r is rational implies r in {((PI / 2) + ((2 * PI) * i)),(((2 * PI) / 3) + ((2 * PI) * i)),(PI + ((2 * PI) * i))} )
set a = (2 * PI) * i;
set R = r - ((2 * PI) * i);
assume ( (PI / 2) + ((2 * PI) * i) <= r & r <= PI + ((2 * PI) * i) ) ; :: thesis: ( not r / PI is rational or not cos r is rational or r in {((PI / 2) + ((2 * PI) * i)),(((2 * PI) / 3) + ((2 * PI) * i)),(PI + ((2 * PI) * i))} )
then A1: ( ((PI / 2) + ((2 * PI) * i)) - ((2 * PI) * i) <= r - ((2 * PI) * i) & r - ((2 * PI) * i) <= (PI + ((2 * PI) * i)) - ((2 * PI) * i) ) by XREAL_1:9;
assume A2: ( r / PI is rational & cos r is rational ) ; :: thesis: r in {((PI / 2) + ((2 * PI) * i)),(((2 * PI) / 3) + ((2 * PI) * i)),(PI + ((2 * PI) * i))}
((2 * PI) * i) / PI = ((2 * i) * PI) / PI
.= 2 * i by XCMPLX_1:89 ;
then A3: (r - ((2 * PI) * i)) / PI = (r / PI) - (2 * i) ;
r - ((2 * PI) * i) = ((2 * PI) * (- i)) + r ;
then cos r = cos (r - ((2 * PI) * i)) by COMPLEX2:9;
then r - ((2 * PI) * i) in {(PI / 2),((2 * PI) / 3),PI} by A1, A2, A3, Th55;
then ( r - ((2 * PI) * i) = PI / 2 or r - ((2 * PI) * i) = (2 * PI) / 3 or r - ((2 * PI) * i) = PI ) by ENUMSET1:def 1;
hence r in {((PI / 2) + ((2 * PI) * i)),(((2 * PI) / 3) + ((2 * PI) * i)),(PI + ((2 * PI) * i))} by ENUMSET1:def 1; :: thesis: verum