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;
reserve FSeq for FinSequence of Sigma;

theorem Th59:
  dom(P * FSeq) = dom FSeq
proof
  for x being object holds x in dom (P * FSeq) iff x in dom FSeq
  proof
    let x be object;
    thus x in dom (P * FSeq) implies x in dom FSeq by FUNCT_1:11;
    assume
A1: x in dom FSeq;
    then reconsider k=x as Element of NAT;
    FSeq.k in Sigma;
    then FSeq.k in dom P by FUNCT_2:def 1;
    hence thesis by A1,FUNCT_1:11;
  end;
  hence thesis by TARSKI:2;
end;
