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
  i >= 0 & j >= 0 implies i div j >= 0
proof
  assume that
A1: i >= 0 and
A2: j >= 0;
A3: i / j - 1 < [\ i / j /] by Def6;
  i / j - 1 >= 0-1 by A1,A2,XREAL_1:9;
  then - 1 < [\ i / j /] by A3,XXREAL_0:2;
  hence thesis by Th8;
end;
