let a, b be Real; :: thesis: for n being Integer st (n - b) * ((n + 1) - a) > 0 & (a - n) * ((n + 1) - b) > 0 holds
( ((n - b) * ((n + 1) - a)) + ((a - n) * ((n + 1) - b)) = a - b & ((|.(a - n).| * |.(b - n).|) * |.((a - n) - 1).|) * |.((b - n) - 1).| <= (|.(a - b).| ^2) / 4 )
let n be Integer; :: thesis: ( (n - b) * ((n + 1) - a) > 0 & (a - n) * ((n + 1) - b) > 0 implies ( ((n - b) * ((n + 1) - a)) + ((a - n) * ((n + 1) - b)) = a - b & ((|.(a - n).| * |.(b - n).|) * |.((a - n) - 1).|) * |.((b - n) - 1).| <= (|.(a - b).| ^2) / 4 ) )
assume that
A1: (n - b) * ((n + 1) - a) > 0 and
A2: (a - n) * ((n + 1) - b) > 0 ; :: thesis: ( ((n - b) * ((n + 1) - a)) + ((a - n) * ((n + 1) - b)) = a - b & ((|.(a - n).| * |.(b - n).|) * |.((a - n) - 1).|) * |.((b - n) - 1).| <= (|.(a - b).| ^2) / 4 )
set s = (n - b) * ((n + 1) - a);
set t = (a - n) * ((n + 1) - b);
A3: sqrt (((n - b) * ((n + 1) - a)) * ((a - n) * ((n + 1) - b))) <= (((n - b) * ((n + 1) - a)) + ((a - n) * ((n + 1) - b))) / 2 by A1, A2, SERIES_3:2;
A4: (sqrt (((n - b) * ((n + 1) - a)) * ((a - n) * ((n + 1) - b)))) ^2 = ((n - b) * ((n + 1) - a)) * ((a - n) * ((n + 1) - b)) by A1, A2, SQUARE_1:def 2;
A5: sqrt (((n - b) * ((n + 1) - a)) * ((a - n) * ((n + 1) - b))) >= 0 by A1, A2, SQUARE_1:def 2;
A6: (n - b) * ((n + 1) - a) = |.((n - b) * ((n + 1) - a)).| by A1, COMPLEX1:43
.= |.(n - b).| * |.((n + 1) - a).| by COMPLEX1:65 ;
A7: (a - n) * ((n + 1) - b) = |.((a - n) * ((n + 1) - b)).| by A2, COMPLEX1:43
.= |.(a - n).| * |.((n + 1) - b).| by COMPLEX1:65 ;
A9: |.(n - b).| = |.(- (n - b)).| by COMPLEX1:52
.= |.(b - n).| ;
A10: |.((n + 1) - a).| = |.(- ((n + 1) - a)).| by COMPLEX1:52
.= |.((a - n) - 1).| ;
A11: |.((n + 1) - b).| = |.(- ((n + 1) - b)).| by COMPLEX1:52
.= |.((b - n) - 1).| ;
A12: ((n - b) * ((n + 1) - a)) * ((a - n) * ((n + 1) - b)) = (|.(n - b).| * |.((n + 1) - a).|) * (|.(a - n).| * |.((n + 1) - b).|) by A7, A6
.= ((|.(a - n).| * |.(b - n).|) * |.((a - n) - 1).|) * |.((b - n) - 1).| by A11, A10, A9 ;
((a - b) / 2) ^2 = ((a - b) ^2) / (2 ^2) by XCMPLX_1:76
.= (|.(a - b).| ^2) / 4 by COMPLEX1:75 ;
hence ( ((n - b) * ((n + 1) - a)) + ((a - n) * ((n + 1) - b)) = a - b & ((|.(a - n).| * |.(b - n).|) * |.((a - n) - 1).|) * |.((b - n) - 1).| <= (|.(a - b).| ^2) / 4 ) by A3, A4, A5, A12, SQUARE_1:15; :: thesis: verum