 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 Lm3:
  (Partial_Union Equal_Div_interval(t)).i = [.0, (i+1)/t .[
  proof
  defpred P[Nat] means
  (Partial_Union Equal_Div_interval(t)).$1 = [.0, ($1 +1)/t .[;
A1: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
A2:   (Partial_Union Equal_Div_interval(t)).k = [.0, (k+1)/t .[;
      (Partial_Union Equal_Div_interval(t)).(k+1) =
      (Partial_Union Equal_Div_interval(t)).k \/ (Equal_Div_interval(t)).(k+1)
      by PROB_3:def 2 .=
      [.0, (k+1)/t .[ \/ [. (k+1)/t, (k+1)/t + t" .[ by Def1,A2
        .= [. 0, (k+2)/t .[ by Lm2;
      hence thesis;
    end;
    (Partial_Union Equal_Div_interval(t)).0 = (Equal_Div_interval(t)).0
      by PROB_3:def 2
      .= [. 0/t,0/t + t" .[ by Def1
      .= [. 0 ,t" .[; then
A3: P[0];
    for n be Nat holds P[n] from NAT_1:sch 2(A3,A1);
    hence thesis;
  end;
