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
  for r being Real st r>=0 holds [/ r \]>=0 & [\ r /]>=0 &
  [/ r \] is Element of NAT & [\ r /] is Element of NAT
proof
  let r be Real;
  assume
A1: r>=0;
A2: r<=[/ r \] by Def7;
  r-1<[\ r /] by Def6;
  then r-1+1<[\ r /]+1 by XREAL_1:6;
  then 0-1<[\ r /]+1-1 by A1,XREAL_1:9;
  then [\ r /]>=0 by Th8;
  hence thesis by A1,A2,Th3;
end;
