let r be Real; :: thesis: for i being Integer st (2 * PI) * i <= r & r <= (PI / 2) + ((2 * PI) * i) & r / PI is rational & sin r is rational holds
r in {((2 * PI) * i),((PI / 6) + ((2 * PI) * i)),((PI / 2) + ((2 * PI) * i))}
let i be Integer; :: thesis: ( (2 * PI) * i <= r & r <= (PI / 2) + ((2 * PI) * i) & r / PI is rational & sin r is rational implies r in {((2 * PI) * i),((PI / 6) + ((2 * PI) * i)),((PI / 2) + ((2 * PI) * i))} )
set a = (2 * PI) * i;
set R = r - ((2 * PI) * i);
assume ( (2 * PI) * i <= r & r <= (PI / 2) + ((2 * PI) * i) ) ; :: thesis: ( not r / PI is rational or not sin r is rational or r in {((2 * PI) * i),((PI / 6) + ((2 * PI) * i)),((PI / 2) + ((2 * PI) * i))} )
then A1: ( ((2 * PI) * i) - ((2 * PI) * i) <= r - ((2 * PI) * i) & r - ((2 * PI) * i) <= ((PI / 2) + ((2 * PI) * i)) - ((2 * PI) * i) ) by XREAL_1:9;
assume A2: ( r / PI is rational & sin r is rational ) ; :: thesis: r in {((2 * PI) * i),((PI / 6) + ((2 * PI) * i)),((PI / 2) + ((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 sin r = sin (r - ((2 * PI) * i)) by COMPLEX2:8;
then r - ((2 * PI) * i) in {0,(PI / 6),(PI / 2)} by A1, A2, A3, Th62;
then ( r - ((2 * PI) * i) = 0 or r - ((2 * PI) * i) = PI / 6 or r - ((2 * PI) * i) = PI / 2 ) by ENUMSET1:def 1;
hence r in {((2 * PI) * i),((PI / 6) + ((2 * PI) * i)),((PI / 2) + ((2 * PI) * i))} by ENUMSET1:def 1; :: thesis: verum