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 Th42:
  ASeq is disjoint_valued implies for n,m st n < m holds (
  Partial_Union ASeq).n misses ASeq.m
proof
  assume
A1: ASeq is disjoint_valued;
  let n,m such that
A2: n < m;
  assume (Partial_Union ASeq).n meets ASeq.m;
  then consider x being object such that
A3: x in (Partial_Union ASeq).n and
A4: x in ASeq.m by XBOOLE_0:3;
  reconsider ASeq as SetSequence of Omega;
  consider k being Nat such that
A5: k <= n and
A6: x in ASeq.k by A3,Th13;
  ASeq.k misses ASeq.m by A1,A2,A5;
  then ASeq.k /\ ASeq.m = {};
  hence contradiction by A4,A6,XBOOLE_0:def 4;
end;
