 reserve i,j,k,m,n,m1,n1 for Nat;
 reserve a,r,r1,r2 for Real;
 reserve m0,cn,cd for Integer;
 reserve x1,x2,o for object;
 reserve t for 1_greater Nat;

theorem Lm4:
  for r be Real, i be Nat st [\ r*t /] = i holds
    r in (Equal_Div_interval(t)).i
  proof
    let r be Real;
    let i be Nat;
    assume [\ r*t /] = i; then
A3: i <= r*t & r*t -1 < i by INT_1:def 6; then
    i/t <= r*t/t by XREAL_1:64; then
    i*t" <= r*(t*t"); then
A6: i*t" <= r*1 by Lm1;
A7: r*t-1 + 1 < i + 1 by A3,XREAL_1:8;
    0 < t" by Lm1; then
    (r*t)*t" < (i+1)*t" by A7,XREAL_1:68;then
    r*(t*t") < (i+1)*t"; then
    r*1 < (i+1)/t by Lm1; then
    r in [. i/t, i/t + t" .[ by A6,XXREAL_1:3;
    hence thesis by Def1;
  end;
