let r be real number ; :: thesis: for i being integer number st (2 * PI ) * i < r & r < (2 * PI ) + ((2 * PI ) * i) holds
cos r < 1
let i be integer number ; :: thesis: ( (2 * PI ) * i < r & r < (2 * PI ) + ((2 * PI ) * i) implies cos r < 1 )
assume that
A1: H₁(i) < r and
A2: r < (2 * PI ) + H₁(i) and
A3: cos r >= 1 ; :: thesis: contradiction
A4: ( H₁(i) - H₁(i) < r - H₁(i) & r - H₁(i) < ((2 * PI ) + H₁(i)) - H₁(i) ) by A1, A2, XREAL_1:11;
A5: cos r <= 1 by Th6;
cos (r - H₁(i)) = cos (r + H₁( - i))
.= cos r by COMPLEX2:10
.= 1 by A3, A5, XXREAL_0:1 ;
hence contradiction by A4, Th34; :: thesis: verum