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 Th15:
  Union Partial_Union A1 = Union A1
proof
  thus Union Partial_Union A1 c= Union A1
  proof
    let x be object;
    assume x in Union Partial_Union A1;
    then consider n being Nat such that
A1: x in (Partial_Union A1).n by PROB_1:12;
    consider k such that
    k <= n and
A2: x in A1.k by A1,Th13;
    thus thesis by A2,PROB_1:12;
  end;
  let x be object;
  assume x in Union A1;
  then consider n being Nat such that
A3: x in A1.n by PROB_1:12;
  x in (Partial_Union A1).n by A3,Th13;
  hence thesis by PROB_1:12;
end;
