let r be Real; :: thesis: for i being Integer st (2 * PI) * i < r & r < (2 * PI) + ((2 * PI) * i) holds
cos r < 1
let i be Integer; :: thesis: ( (2 * PI) * i < r & r < (2 * PI) + ((2 * PI) * i) implies cos r < 1 )
assume that
A1: ( H₁(i) < r & r < (2 * PI) + H₁(i) ) and
A2: cos r >= 1 ; :: thesis: contradiction
A3: ( H₁(i) - H₁(i) < r - H₁(i) & r - H₁(i) < ((2 * PI) + H₁(i)) - H₁(i) ) by A1, XREAL_1:9;
A4: cos r <= 1 by Th6;
cos (r - H₁(i)) = cos (r + H₁( - i))
.= cos r by COMPLEX2:9
.= 1 by A2, A4, XXREAL_0:1 ;
hence contradiction by A3, Th34; :: thesis: verum