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 Th37:
  |.a-[\a/].|*|.b-[\a/].| >= |.a-b.|/2 &
    |.a-([\a/]+1).|*|.b-([\a/]+1).| >= |.a-b.|/2 implies
  a is Integer or [\a/]+1 >= b
  proof
    set u=[\a/],v=u+1;
    assume
AA: |.a-[\a/].|*|.b-[\a/].| >= |.a-b.|/2 &
    |.a-([\a/]+1).|*|.b-([\a/]+1).| >= |.a-b.|/2;
    assume
A2: a is not Integer & [\a/]+1 < b; then
A3: a - [\a/] > 0 by INT_1:26,XREAL_1:50;
A4: a - 1 < [\a/] by INT_1:def 6; then
a3: a < [\a/] + 1 by XREAL_1:19; then
A5: |.a-[\a/].|*|.b-[\a/].|*|.a-[\a/]-1.|*|.b-[\a/]-1.| <= |.a - b.|^2/4
      by A2,INT_1:26,Th27;
    per cases by XXREAL_0:1,AA;
    suppose
A6:   |.a-[\a/].|*|.b-[\a/].| = |.a-b.|/2 &
      |.a-([\a/]+1).|*|.b-([\a/]+1).| > |.a-b.|/2;
      (1+[\a/])-[\a/] < b-[\a/] by A2,XREAL_1:14; then
      (|.a-[\a/].|*|.b-[\a/].|)*(|.a-([\a/]+1).|*|.b-([\a/]+1).|)
        > (|.a-b.|/2)*(|.a-b.|/2) by A6,A3,XREAL_1:98;
      hence thesis by A5;
    end;
    suppose
A6:   |.a-[\a/].|*|.b-[\a/].| > |.a-b.|/2 &
      |.a-([\a/]+1).|*|.b-([\a/]+1).| = |.a-b.|/2;
a6:  a-([\a/]+1) < 0 by a3,XREAL_1:49;
     b-([\a/]+1) > 0 by A2,XREAL_1:50; then
     (|.a-[\a/].|*|.b-[\a/].|)*(|.a-([\a/]+1).|*|.b-([\a/]+1).|)
     > (|.a-b.|/2)*(|.a-b.|/2) by a6,XREAL_1:98,A6;
     hence thesis by A5;
    end;
    suppose |.a-[\a/].|*|.b-[\a/].| > |.a-b.|/2 &
        |.a-([\a/]+1).|*|.b-([\a/]+1).| > |.a-b.|/2;
     hence thesis by Th23,A5;
    end;
    suppose
SS:   |.a-([\a/]+1).|*|.b-([\a/]+1).| = |.a-b.|/2 &
      |.a-[\a/].|*|.b-[\a/].| = |.a-b.|/2;
A6:   [\a/]+1-b < 0 by A2,XREAL_1:49;
A7:   [\a/] - a < 0 by A2,INT_1:26,XREAL_1:49;
A8:   (1+[\a/])-[\a/] < b-[\a/] by A2,XREAL_1:14;
A9:   [\a/] + 1-a > 0 by A4,XREAL_1:19,XREAL_1:50;
A11:  (u- a)*(u+ 1- b) = |.(u- a)*(u+ 1- b).| by A6,A7,ABSVALUE:def 1
        .= |.(u- a).| * |.(u+ 1- b).| by COMPLEX1:65
        .= |.-(u- a).| * |.u+ 1- b.| by COMPLEX1:52 .= |.a-u.| * |.u+ 1- b.|;
      (b-u)*(u+ 1- a) = |.(b-u)*(u+ 1- a).| by A8,A9,ABSVALUE:def 1
        .= |.b-u.| * |.u+ 1-a.| by COMPLEX1:65; then
A12:  |.a-u.|*|.u+1-b.|+|.b-u.|*|.u+1-a.| = |.b-a.| by A11
        .= |.-(b-a).| by COMPLEX1:52 .=|.a-b.|;
      set r1 = |.a-u.|*|.u+1-b.|;
      set r2 = |.b-u.|*|.u+1-a.|;
A16:  |.u+1-a.| = |.-(u+1-a).| by COMPLEX1:52 .= |.a-u-1.|;
A17:  |.u+1-b.| = |.-(u+1-b).| by COMPLEX1:52 .= |.b-u-1.|;
      |.a-u.|*|.b-u.|*|.a-u-1.|*|.b-u-1.|
         = (|.a-u.|*|.b-u.|)*(|.a-u-1.|*|.b-u-1.|)
        .= ((r1+r2)/2)^2 by A12,SS; then
      r1*r2 = ((r1+r2)/2)^2 by A17,A16; then
A18:  r1 = r2 by Th20;
A22:  |.u+1-b.|-|.b-u.|
    = (|.u+1-b.|*|.a-u.|-|.b-u.|*|.a-u.|)/|.a-u.| by A3,XCMPLX_1:129
   .= 0 by A12,A18,SS;
       |.u+1-b.|-|.b-u.| = -(u+1-b) - |.b-u.| by A2,XREAL_1:49,ABSVALUE:def 1
   .= -(u+1-b) - (b-u) by A8,ABSVALUE:def 1 .= -1;
   hence contradiction by A22;
    end;
  end;
