theorem Th52:
 for n being Nat holds
  B is non-ascending implies (inferior_setsequence(B)).n = Intersection B
proof let n be Nat;
  defpred P[Nat] means (inferior_setsequence(B)).$1 = Intersection B;
  assume B is non-ascending;
  then
A1: for k being Nat st P[k] holds P[k+1] by Th51;
A2: P[0] by Th17;
  for k being Nat holds P[k] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
