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
  0 < i & 1 < j implies i div j < i
proof
  assume that
A1: 0 < i and
A2: 1 < j;
  assume
A3: i <= i div j;
  i div j <= i/j by Def6;
  then j * (i div j) <= j * (i/j) by A2,XREAL_1:64;
  then j * (i div j) <= i by A2,XCMPLX_1:87;
  then j * (i div j) <= i div j by A3,XXREAL_0:2;
  then j * (i div j) * (i div j)" <= (i div j) * (i div j)" by A1,A3,XREAL_1:64
;
  then j * ((i div j) * (i div j)") <= (i div j) * (i div j)";
  then j * 1 <= (i div j) * (i div j)" by A1,A3,XCMPLX_0:def 7;
  hence contradiction by A1,A2,A3,XCMPLX_0:def 7;
end;
