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;
reserve r, s for Real;
reserve i for Integer,
  a, b, r, s for Real;

theorem
  for m,n being natural Number holds
  m^2 <= n < (m+1)^2 implies [\ sqrt n /] = m
  proof
    let m,n be natural Number;
    assume m^2 <= n;
    then
A1: sqrt (m^2) <= sqrt n by SQUARE_1:26;
    assume n < (m+1)^2;
    then
A2: sqrt n < sqrt ((m+1)^2) by SQUARE_1:27;
A3: sqrt (m^2) = m by SQUARE_1:def 2;
    sqrt ((m+1)^2) = m+1 by SQUARE_1:def 2;
    then sqrt n - 1 < m+1-1 by A2,XREAL_1:14;
    hence thesis by A1,A3,Def6;
  end;
