let r be real number ; :: thesis: ( cos r = 1 implies r = (2 * PI ) * [\(r / (2 * PI ))/] )
set i = [\(r / (2 * PI ))/];
assume A1: cos r = 1 ; :: thesis: r = (2 * PI ) * [\(r / (2 * PI ))/]
consider w being real number such that
A2: w = ((2 * PI ) * (- [\(r / (2 * PI ))/])) + r and
A3: 0 <= w and
A4: w < 2 * PI by COMPLEX2:2;
A5: 0 + H₁([\(r / (2 * PI ))/]) <= w + H₁([\(r / (2 * PI ))/]) by A3, XREAL_1:8;
A6: w + H₁([\(r / (2 * PI ))/]) < (2 * PI ) + H₁([\(r / (2 * PI ))/]) by A4, XREAL_1:8;
((sin r) * (sin r)) + (1 * 1) = 1 by A1, SIN_COS:32;
then sin r = 0 ;
then ( r = 0 + H₁([\(r / (2 * PI ))/]) or r = PI + H₁([\(r / (2 * PI ))/]) ) by A2, A5, A6, Th21;
hence r = (2 * PI ) * [\(r / (2 * PI ))/] by A1, COMPLEX2:10, SIN_COS:82; :: thesis: verum