reserve X for set, x,y,z for object,
  k,l,n for Nat,
  r for Real;
reserve i,i0,i1,i2,i3,i4,i5,i8,i9,j for Integer;
reserve r1,p,p1,g,g1,g2 for Real,
  Y for Subset of REAL;

theorem
  i2 > 0 implies i1 mod i2 < i2
proof
  assume
A1: i2 > 0;
  i1/i2 -1 < [\ i1/i2 /] by Def6;
  then (i1/i2 -1)*i2 < (i1 div i2)*i2 by A1,XREAL_1:68;
  then i1/i2*i2 -1*i2 < (i1 div i2)*i2;
  then i1 -i2 < (i1 div i2)*i2-0 by A1,XCMPLX_1:87;
  then i1 -(i1 div i2)*i2 < i2-0 by XREAL_1:17;
  hence thesis by A1,Def10;
end;
