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 Th9:
  A1.n c= (Partial_Union A1).n
proof
  per cases by NAT_1:6;
  suppose
    n = 0;
    hence thesis by Def2;
  end;
  suppose
    ex k being Nat st n = k+1;
    then consider k such that
A1: n = k+1;
    (Partial_Union A1).(k+1) = (Partial_Union A1).k \/ A1.(k+1) by Def2;
    hence thesis by A1,XBOOLE_1:7;
  end;
end;
