reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem Th5:
for h be PartFunc of REAL,the carrier of S,
    seq be Real_Sequence
 st rng seq c= dom h
  holds ||.h/*seq .|| = ||.h.||/*seq
      & -(h/*seq) = (-h)/*seq
proof
   let h be PartFunc of REAL,the carrier of S,
       seq be Real_Sequence;
   assume A1: rng seq c= dom h; then
A2:rng seq c= dom ||.h.|| by NORMSP_0:def 3;
   now
    let n be Element of NAT;
    seq.n in rng seq by FUNCT_2:4; then
    seq.n in dom h by A1; then
A3: seq.n in dom ||.h.|| by NORMSP_0:def 3;
    thus ||.h/*seq.||.n = ||.(h/*seq).n.|| by NORMSP_0:def 4
      .= ||.h/.(seq.n).|| by A1,FUNCT_2:109
      .= ||.h.||.(seq.n) by A3,NORMSP_0:def 3
      .= ||.h.||/.(seq.n) by A3,PARTFUN1:def 6
      .= (||.h.||/*seq).n by A2,FUNCT_2:109;
   end;
   hence ||.h/*seq.|| = (||.h.||)/*seq by FUNCT_2:63;
   now let n be Element of NAT;
    thus (-(h/*seq)).n = - ((h/*seq).n) by BHSP_1:44
     .= (-1) * (h/*seq).n by RLVECT_1:16
     .= ((-1)*(h/*seq)).n by NORMSP_1:def 5
     .= (((-1)(#)h)/*seq).n by A1,Th4
     .= ((-h)/*seq).n by VFUNCT_1:23;
   end;
   hence -(h/*seq) = (-h)/*seq by FUNCT_2:63;
end;
