let r be real number ; :: thesis: ( sin r = - 1 implies r = ((3 / 2) * PI ) + ((2 * PI ) * [\(r / (2 * PI ))/]) )
set i = [\(r / (2 * PI ))/];
consider w being real number such that
A1: w = ((2 * PI ) * (- [\(r / (2 * PI ))/])) + r and
A2: ( 0 <= w & w < 2 * PI ) by COMPLEX2:2;
assume A3: sin r = - 1 ; :: thesis: r = ((3 / 2) * PI ) + ((2 * PI ) * [\(r / (2 * PI ))/])
then ((cos r) * (cos r)) + ((- 1) * (- 1)) = 1 by SIN_COS:32;
then A4: cos r = 0 ;
( 0 + H₁([\(r / (2 * PI ))/]) <= w + H₁([\(r / (2 * PI ))/]) & w + H₁([\(r / (2 * PI ))/]) < (2 * PI ) + H₁([\(r / (2 * PI ))/]) ) by A2, XREAL_1:8;
then ( r = (PI / 2) + H₁([\(r / (2 * PI ))/]) or r = ((3 / 2) * PI ) + H₁([\(r / (2 * PI ))/]) ) by A1, A4, Th22;
hence r = ((3 / 2) * PI ) + ((2 * PI ) * [\(r / (2 * PI ))/]) by A3, COMPLEX2:9, SIN_COS:82; :: thesis: verum