 reserve Omega for non empty set;
 reserve r for Real;
 reserve Sigma for SigmaField of Omega;
 reserve P for Probability of Sigma;

theorem Th14:
  for DX be non empty set, F be Function of DX,REAL,
  Y1,Y2 be finite Subset of DX st
  Y1 c= Y2 & for x be set st x in Y2 holds 0<= F.x holds
  setopfunc(Y1,DX,REAL,F,addreal)
  <= setopfunc(Y2,DX,REAL,F,addreal)
  proof
    let DX be non empty set, F be Function of DX,REAL,
    Y1,Y2 be finite Subset of DX;
    assume A1:
    Y1 c= Y2 & for x be set st x in Y2 holds 0<= F.x;
    consider p1 being FinSequence of DX such that
    A2: p1 is one-to-one & rng p1 = Y1 & setopfunc(Y1,DX,REAL,F,addreal)
    =Sum(Func_Seq(F,p1)) by Th9;
    reconsider Y3 = Y2 \ Y1 as finite Subset of DX;
    consider p2 being FinSequence of DX such that
    A3: p2 is one-to-one & rng p2 = Y3 &
    setopfunc(Y3,DX,REAL,F,addreal)=Sum(Func_Seq(F,p2)) by Th9;
    now let i be Nat;
      assume A4: i in dom (Func_Seq(F,p2)); then
      A5: (Func_Seq(F,p2)).i = F.(p2.i) by FUNCT_1:12;
      i in dom p2 by A4,FUNCT_1:11; then
      A6: p2.i in Y3 by A3,FUNCT_1:3;
      Y3 c= Y2 by XBOOLE_1:36;
      hence 0 <= (Func_Seq(F,p2)).i by A5,A1,A6;
    end; then
    A7: 0 <= Sum(Func_Seq(F,p2)) by RVSUM_1:84;
    reconsider p3=p1^p2 as FinSequence of DX;
    A8: rng p3 = rng p1 \/ rng p2 by FINSEQ_1:31
    .=Y1 \/ Y2 by A3,A2,XBOOLE_1:39
    .=Y2 by A1,XBOOLE_1:12;
    rng p1 misses rng p2 by A2,A3,XBOOLE_1:79;
    then
     p3 is one-to-one by A2,A3,FINSEQ_3:91;
    then A9: setopfunc(Y2,DX,REAL,F,addreal)
    =Sum(Func_Seq(F,p3)) by A8,Th10;
    A10: Func_Seq(F,p3)= Func_Seq(F,p1) ^ Func_Seq(F,p2) by Lm5;
    Sum(Func_Seq(F,p1)) + (0 qua Real)
    <= Sum(Func_Seq(F,p1)) + Sum(Func_Seq(F,p2)) by A7,XREAL_1:6;
    hence thesis by A2,A10,A9,RVSUM_1:75;
  end;
