
theorem Th37:
  for S be non empty finite set,
  X be Subset of S,
  s be FinSequence of S holds
  s"X = trueEVENT((MembershipDecision(X))*s)
  proof
    let S be non empty finite set,
    X be Subset of S,
    s be FinSequence of S;
    set SX= s"X;
    reconsider SX as Subset of (dom s) by RELAT_1:132;
    dom ((MembershipDecision(X))*s) c= dom s by RELAT_1:25;then
    reconsider SY= trueEVENT((MembershipDecision(X))*s)
    as Subset of dom s by XBOOLE_1:1;
    consider f be Function of S,BOOLEAN such that
    A1: f=chi(X,S) & MembershipDecision(X) =f;
    A2:dom f = S by A1,FUNCT_3:def 3;
    A3:
    for x be object st x in SY holds x in SX
    proof
      let x be object;
      assume A4:x in SY;
      s.x in trueEVENT( f) by A1,Th13,A4;
      then s.x in dom f &
      f.(s.x) in {TRUE} by FUNCT_1:def 7;
      then s.x in dom f & f.(s.x) = TRUE by TARSKI:def 1;then
      s.x in X by A1,FUNCT_3:36;
      hence thesis by A4,FUNCT_1:def 7;
    end;
    for x be object st x in SX holds x in SY
    proof
      let x be object;
      assume A5:x in SX;
      A6:s.x in rng s by A5,FUNCT_1:3;
      s.x in X by A5,FUNCT_1:def 7;then
      f.(s.x) = TRUE by A1,FUNCT_3:def 3;then
      f.(s.x) in {TRUE} by TARSKI:def 1;then
      s.x in trueEVENT(f) by A6,A2,FUNCT_1:def 7;
      hence thesis by A1,Th13,A5;
    end;
    hence thesis by A3,TARSKI:2;
  end;
