let a, b be Real; 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; ( (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
; ( ((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; verum