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 Th52:
  for i being Integer, r being Real st i <= r holds i <= [\ r /]
proof
  let i be Integer;
  let r be Real;
  assume i <= r;
  then
A1: i-1 <= r-1 by XREAL_1:9;
  r-1 < [\ r /] by Def6;
  then i-1 < [\ r /] by A1,XXREAL_0:2;
  then i-1+1 <= [\ r /] by Th7;
  hence thesis;
end;
