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 Th5:
  for h1,h2 be PartFunc of REAL, the carrier of F
  for seq be Real_Sequence st rng seq c= dom h1 /\ dom h2 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;
    A1: dom h1 /\ dom h2 c= dom h1 by XBOOLE_1:17;
    A2: dom h1 /\ dom h2 c= dom h2 by XBOOLE_1:17;
    assume
    A3: rng seq c= dom h1 /\ dom h2; then
    A4: rng seq c= dom (h1 + h2) by VFUNCT_1:def 1;
    now let n be Nat;
A5:   n in NAT by ORDINAL1:def 12;
      A6:seq.n in rng seq by FUNCT_2:4,A5;
      thus ((h1+h2)/*seq).n = (h1+h2)/.(seq.n) by A4,FUNCT_2:109,A5
      .= h1/.(seq.n) + h2/.(seq.n) by A4,A6,VFUNCT_1:def 1
      .= (h1/*seq).n + h2/.(seq.n) by A3,A1,FUNCT_2:109,XBOOLE_1:1,A5
      .= (h1/*seq).n + (h2/*seq).n by A3,A2,FUNCT_2:109,XBOOLE_1:1,A5;
    end;
    hence (h1+h2)/*seq=h1/*seq+h2/*seq by NORMSP_1:def 2;
    A7: rng seq c= dom (h1 - h2) by A3,VFUNCT_1:def 2;
    now
      let n be Nat;
A8:   n in NAT by ORDINAL1:def 12;
      A9:seq.n in rng seq by FUNCT_2:4,A8;
      thus ((h1-h2)/*seq).n = (h1-h2)/.(seq.n) by A7,FUNCT_2:109,A8
      .= h1/.(seq.n) - h2/.(seq.n) by A7,A9,VFUNCT_1:def 2
      .= (h1/*seq).n - h2/.(seq.n) by A3,A1,FUNCT_2:109,XBOOLE_1:1,A8
      .= (h1/*seq).n - (h2/*seq).n by A3,A2,FUNCT_2:109,XBOOLE_1:1,A8;
    end;
    hence thesis by NORMSP_1:def 3;
  end;
