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

theorem Th14:
  for h be PartFunc of S,T for seq be sequence of S st rng seq c=
  dom h holds ||.h/*seq .|| = ||.h.||/*seq & -(h/*seq) = (-h)/*seq
proof
  let h be PartFunc of S,T;
  let seq be sequence of S;
  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 Th6;
    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;
  thus -(h/*seq) =(-jj)*(h/*seq) by Th2
    .= ((-1)(#)h)/*seq by A1,Th13
    .= (-h)/*seq by VFUNCT_1:23;
end;
