reserve a,a1,a2,v,v1,v2,x for object;
reserve V,A for set;
reserve m,n for Nat;
reserve S,S1,S2 for FinSequence;
reserve D,D1,D2 for NonatomicND of V,A;

theorem Th58:
  for f being V-valued FinSequence holds
  1 <= n <= len f implies
  namingSeq(V,A,f,a).n is NonatomicND of V,A
  proof
    let f be V-valued FinSequence;
    assume that
A1: 1 <= n and
A2: n <= len f;
    set g = namingSeq(V,A,f,a);
    per cases by A1,XXREAL_0:1;
    suppose n = 1;
      then g.n = naming(V,A,f.len f,a) by A2,Def14;
      hence thesis;
    end;
    suppose
A3:   n > 1;
      then reconsider k = n-1 as Element of NAT by INT_1:5;
      1-1 < k by A3,XREAL_1:9;
      then
A4:   0+1 <= k by INT_1:7;
A5:   len f = len g by A1,A2,Def14;
      k+0 < k+1 by XREAL_1:8;
      then k < len g by A2,A5,XXREAL_0:2;
      then g.(k+1) = naming(V,A,f.(len f-k),g.k) by A2,A4,Def14;
      hence thesis;
    end;
  end;
