reserve n,m,k for Nat;
reserve r,r1 for Real;
reserve f,seq,seq1 for Real_Sequence;
reserve x,y for set;
reserve e1,e2 for ExtReal;
reserve Nseq for increasing sequence of NAT;

theorem
  for n holds n<=Nseq.n
proof
  defpred X[Nat] means $1<=Nseq.$1;
A1: now
    let k such that
A2: X[k];
    Nseq.k<Nseq.(k+1) by Lm7;
    then k<Nseq.(k+1) by A2,XXREAL_0:2;
    hence X[k+1] by NAT_1:13;
  end;
A3: X[0];
  thus for k holds X[k] from NAT_1:sch 2(A3,A1);
end;
