 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 Lm5:
  r1 in (Equal_Div_interval(t)).i & r2 in (Equal_Div_interval(t)).i
  implies |. r1 - r2 .| < t"
  proof
    assume that
A2: r1 in (Equal_Div_interval(t)).i and
A3: r2 in (Equal_Div_interval(t)).i;
A5: r1 in [. i/t, i/t + t" .[ &
    r1 in [. i/t, i/t + t" .[ & r2 in [. i/t, i/t + t" .[ by A2,A3,Def1;
    then
A6: i/t <= r1 & i/t <= r2 & r1 < i/t +t" & r2 < i/t + t" by XXREAL_1:3;
    per cases by XXREAL_0:1;
      suppose r1=r2; then
        |. r1 - r2 .| = 0 by ABSVALUE:2;
        hence thesis by Lm1;
      end;
      suppose r1 > r2; then
A9:     r1 - r2 > 0 by XREAL_1:50;
        i*t" + t" - i*t" > r1 - r2 by A6,XREAL_1:14;
        hence thesis by A9,ABSVALUE:def 1;
      end;
      suppose
A10:    r2 > r1;
        r2< i*t" + t" by A5,XXREAL_1:3; then
A12:    i*t" + t" - r1  > r2 - r1 by XREAL_1:14;
        i*t" <= r1 by A5,XXREAL_1:3; then
A13:    i*t" + t" - i*t" >= i*t" + t" - r1 by XREAL_1:13;
        |. r1 - r2 .| = -(r1 - r2) by A10,XREAL_1:49,ABSVALUE:def 1;
        hence thesis by A13,A12,XXREAL_0:2;
      end;
  end;
