reserve r1,r2,r3 for non negative Real;
reserve n,m1 for Nat;
reserve s for Real;
reserve cn,cd,i1,j1 for Integer;
reserve r for irrational Real;
reserve q for Rational;
reserve c0,c1,c2,u,a0,b0 for Real;
reserve a,b for Real;
reserve n for Integer;

theorem Th24:
   (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
   proof
     assume that
A1:  (n-b)*(n+1-a) > 0 and
A2:  (a-n)*(n+1-b) > 0;
     set s=(n-b)*(n+1-a);
     set t=(a-n)*(n+1-b);
A3:  sqrt(s*t)<=(s+t)/2 by A1,A2,SERIES_3:2;
A4:  (sqrt(s*t))^2=s*t by A1, A2, SQUARE_1:def 2;
A5:  sqrt(s*t) >= 0 by A1,A2,SQUARE_1:def 2;
A6:  s=|.s.| by A1,COMPLEX1:43
     .=|.n-b.|*|.n+1-a.| by COMPLEX1:65;
A7:  t=|.t.| 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: s*t = |.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 thesis by A3,A4,A5,A12,SQUARE_1:15;
   end;
