 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 Lm7:
  for r be Real st r in [.0,1.[ holds
    ex i be Nat st i <= t-1 & r in (Equal_Div_interval(t)).i
  proof
    let r be Real;
    assume r in [.0,1.[; then
A2: r in (Partial_Union Equal_Div_interval(t)).(t-1) by Lm6;
A3: t-1 > 0 by Lm1,XREAL_1:50;
    t -1 in NAT by A3,INT_1:3;
    hence thesis by A2,PROB_3:13;
  end;
