reserve m,n for non zero Element of NAT;
reserve i,j,k for Element of NAT;
reserve Z for set;

theorem Th19:
for f be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS 1,REAL-NS j)
  ex p be Point of REAL-NS j st p = f.<*1*> & ||.p.|| = ||.f.||
proof
   let f be Point of
        R_NormSpace_of_BoundedLinearOperators(REAL-NS 1,REAL-NS j);
   reconsider g = f as Lipschitzian LinearOperator of REAL-NS 1,REAL-NS j
     by LOPBAN_1:def 9;
   consider p be Point of REAL-NS j such that
A1: p = f.<*1*>
  & (for r be Real, x be Point of REAL-NS 1 st x = <*r*> holds f.x = r*p)
  & (for x be Point of REAL-NS 1 holds ||. f.x .|| = ||.p.|| * ||.x.||)
       by Th18;
   <*jj*> in REAL 1 by FINSEQ_2:98; then
   reconsider One = <*jj*> as Point of REAL-NS 1 by REAL_NS1:def 4;
   ||. g.One .|| <= ||.f.|| * ||.One.|| by LOPBAN_1:32; then
   ||. g.One .|| <= ||.f.|| * |.1.| by PDIFF_8:2; then
A2: ||. f.One .|| <= ||.f.|| * 1 by ABSVALUE:def 1;
   for x be Point of REAL-NS 1 st
     ||.x.|| <= 1 holds ||. f.x .|| <= ||.p.|| * ||.x.|| by A1; then
   ||.f.|| <= ||.p.|| by Th1;
   hence thesis by A1,A2,XXREAL_0:1;
end;
