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;
