reserve n,n1,m,m1,k for Nat;
reserve x,X,X1 for set;
reserve g,g1,g2,t,x0,x1,x2 for Complex;
reserve s1,s2,q1,seq,seq1,seq2,seq3 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f,f1,f2,h,h1,h2 for PartFunc of COMPLEX,COMPLEX;
reserve p,r,s for Real;
reserve Ns,Nseq for increasing sequence of NAT;

theorem
  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
  assume h1 is total & h2 is total;
  then dom (h1+h2) = COMPLEX by PARTFUN1:def 2;
  then dom h1 /\ dom h2 = COMPLEX by VALUED_1:def 1;
  then
A1: rng seq c= dom h1 /\ dom h2;
  hence (h1+h2)/*seq = h1/*seq + h2/*seq by Th7;
  thus (h1-h2)/*seq = h1/*seq - h2/*seq by A1,Th7;
  thus thesis by A1,Th7;
end;
