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 Th21:
   a is not Integer & (n=[\a/] or n=[\a/]+1) implies |.a-n.|<1
   proof
     assume
A1:  a is not Integer & (n=[\a/] or n=[\a/]+1);
     per cases by A1;
       suppose
A2:      n=[\a/]; then
A3:      a-n > 0 by A1,INT_1:26,XREAL_1:50;
         a-[\a/] < (1+[\a/])-[\a/] by INT_1:29,XREAL_1:14;
         hence thesis by A2,A3,ABSVALUE:def 1;
       end;
       suppose
A5:      n = [\a/]+1; then
A6:      a - n < 0 by INT_1:29,XREAL_1:49;
         [\a/] < [/a\] by A1,INT_1:35; then
         n = [/a\] by A5,INT_1:41; then
         n < a + 1 by INT_1:def 7; then
A8:      n-a < 1 by XREAL_1:19;
         |.a-n.|=-(a-n) by A6,ABSVALUE:def 1;
         hence thesis by A8;
       end;
     end;
