reserve n,m,k,i for Nat,
  g,s,t,p for Real,
  x,y,z for object, X,Y,Z for set,
  A1 for SetSequence of X,
  F1 for FinSequence of bool X,
  RFin for real-valued FinSequence,
  Si for SigmaField of X,
  XSeq,YSeq for SetSequence of Si,
  Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq,BSeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem
 for x being object holds
  x in (Partial_Intersection XSeq).n iff
    for k st k <= n holds x in XSeq.k
 proof reconsider XSeq as SetSequence of X;
  let x be object;
  x in (Partial_Intersection XSeq).n iff
    for k st k <= n holds x in XSeq.k by Th12;
  hence thesis;
 end;
