reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1 for sequence of S;
reserve x0 for Point of S;
reserve h for (0.S)-convergent sequence of S;
reserve c for constant sequence of S;
reserve R,R1,R2 for RestFunc of S,T;
reserve L,L1,L2 for Point of R_NormSpace_of_BoundedLinearOperators(S,T);

theorem Th25:
  for h1,h2 be PartFunc of S,T for seq be sequence of S holds h1
is total & h2 is total implies (h1+h2)/*seq = h1/*seq + h2/*seq & (h1-h2)/*seq
  = h1/*seq - h2/*seq
proof
  let h1,h2 be PartFunc of S,T;
  let seq be sequence of S;
  assume h1 is total & h2 is total;
  then h1+h2 is total by VFUNCT_1:32;
  then dom (h1+h2) = the carrier of S by PARTFUN1:def 2;
  then dom h1 /\ dom h2 = the carrier of S by VFUNCT_1:def 1;
  then
A1: rng seq c= dom h1 /\ dom h2;
  hence (h1+h2)/*seq = h1/*seq + h2/*seq by NFCONT_1:12;
  thus thesis by A1,NFCONT_1:12;
end;
