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 Th31:
  r < [/ r \] iff r is not integer
proof
  now
    assume
A1: not r is Integer;
    r <= [/ r \] by Def7;
    hence r < [/ r \] by A1,XXREAL_0:1;
  end;
  hence thesis;
end;
