reserve x,X,Y for set;
reserve g,r,r1,r2,p,p1,p2 for Real;
reserve R for Subset of REAL;
reserve seq,seq1,seq2,seq3 for Real_Sequence;
reserve Ns for increasing sequence of NAT;
reserve n for Nat;
reserve W for non empty set;
reserve h,h1,h2 for PartFunc of W,REAL;

theorem
  for h1,h2 being PartFunc of W,REAL, seq being sequence of W holds
  h1 is total & h2 is total implies
  (h1+h2)/*seq = h1/*seq + h2/*seq & (h1-h2)/*seq = h1/*seq - h2/*seq &
  (h1(#)h2)/*seq = (h1/*seq) (#) (h2/*seq)
proof
  let h1,h2 be PartFunc of W,REAL, seq be sequence of W;
  assume h1 is total & h2 is total;
  then dom (h1+h2) = W by PARTFUN1:def 2;
  then dom h1 /\ dom h2 = W by VALUED_1:def 1;
  then
A1: rng seq c= dom h1 /\ dom h2;
  hence (h1+h2)/*seq = h1/*seq + h2/*seq by Th8;
  thus (h1-h2)/*seq = h1/*seq - h2/*seq by A1,Th8;
  thus thesis by A1,Th8;
end;
