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

theorem Th12:
  for DX1,DX2 be non empty set, F1 be Function of DX1,REAL,
  F2 be Function of DX2,REAL,
  G be Function of [:DX1,DX2:], REAL,
  Y1 be non empty finite Subset of DX1,
  Y2 be non empty finite Subset of DX2,
  Y3 be finite Subset of [:DX1,DX2:] st Y3= [:Y1,Y2:] &
  for x,y be set st x in Y1 & y in Y2 holds G.(x,y)= (F1.x)*(F2.y)
  holds
  setopfunc(Y3,[:DX1,DX2:],REAL,G,addreal)
  =setopfunc(Y1,DX1,REAL,F1,addreal) * setopfunc(Y2,DX2,REAL,F2,addreal)
  proof
    let DX1,DX2 be non empty set,
    F1 be Function of DX1,REAL,
    F2 be Function of DX2,REAL,
    G be Function of [:DX1,DX2:],REAL,
    Y1 be non empty finite Subset of DX1,
    Y2 be non empty finite Subset of DX2,
    Y3 be finite Subset of [:DX1,DX2:];
    assume A1:
    Y3= [:Y1,Y2:] & for x,y be set st x in Y1 & y in Y2
    holds G.(x,y)= (F1.x)*(F2.y);
    consider p1 being FinSequence of DX1 such that
    A2: p1 is one-to-one & rng p1 = Y1 & setopfunc(Y1,DX1,REAL,F1,addreal)
    =Sum(Func_Seq(F1,p1)) by Th9;
    consider p2 being FinSequence of DX2 such that
    A3: p2 is one-to-one & rng p2 = Y2 & setopfunc(Y2,DX2,REAL,F2,addreal)
    =Sum(Func_Seq(F2,p2)) by Th9;
    consider p3 being FinSequence of [:DX1,DX2:] such that
    A4: p3 is one-to-one & rng p3 = Y3 &
    setopfunc(Y3,[:DX1,DX2:],REAL,G,addreal)
    =Sum(Func_Seq(G,p3)) by Th9;
    thus thesis by A1,A2,A3,A4,Th11;
  end;
