 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 Lm8:
  for r be Real, i be Nat st r in (Equal_Div_interval(t)).i holds
    [\ r*t /] = i
  proof
   let r be Real;
   let i be Nat;
   assume
A1:  r in (Equal_Div_interval(t)).i;
     r in [. i/t, i/t + t" .[ by A1,Def1; then
A3:  i/t <= r & r < i/t + t" by XXREAL_1:3; then
     i*t"*t <= r*t by XREAL_1:64; then
     i*(t"*t) <= r*t;then
A4:  i*1 <= r*t by Lm1;
     t > 0 by Lm1; then
     r*t < (i + 1)*t" * t by A3,XREAL_1:68; then
     r*t < (i + 1)*(t" * t); then
A6:  r*t < (i + 1)*1 by Lm1;
     i <= r*t & r*t - 1 < i + 1 - 1 by A4,A6,XREAL_1:9;
     hence thesis by INT_1:def 6;
   end;
