reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve X for set;
reserve x,x0,g,r,s,p for Real;
reserve n,m,k for Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,the carrier of F;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve R,R1,R2 for RestFunc of F;
reserve L,L1,L2 for LinearFunc of F;

theorem Th6:
  for h1,h2 be PartFunc of REAL,the carrier of F
  for seq be Real_Sequence st h1 is total & h2 is total holds
  (h1+h2)/*seq = h1/*seq + h2/*seq & (h1-h2)/*seq = h1/*seq - h2/*seq
  proof
    let h1,h2 be PartFunc of REAL,the carrier of F;
    let seq be Real_Sequence;
    assume h1 is total & h2 is total;
    then h1+h2 is total by VFUNCT_1:32;
    then dom (h1+h2) = REAL by PARTFUN1:def 2;
    then dom h1 /\ dom h2 = REAL by VFUNCT_1:def 1;then
A1: rng seq c= dom h1 /\ dom h2;
    hence (h1+h2)/*seq = h1/*seq + h2/*seq by Th5;
    thus thesis by A1,Th5;
  end;
