reserve n,m,k,k1,k2,i,j for Nat;
reserve x,y,z for object,X,Y,Z for set;
reserve A for Subset of X;
reserve B,A1,A2,A3 for SetSequence of X;
reserve Si for SigmaField of X;
reserve S,S1,S2,S3 for SetSequence of Si;

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;
