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

theorem Th20:
for f be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS 1,REAL-NS j),
    x be Point of REAL-NS 1 holds
  ||. f.x .|| = ||.f.|| * ||.x.||
proof
   let f be Point of
           R_NormSpace_of_BoundedLinearOperators(REAL-NS 1,REAL-NS j),
       x be Point of REAL-NS 1;
A1:ex p be Point of REAL-NS j st 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;
   ex q be Point of REAL-NS j st q = f.<*1*> & ||.f.|| = ||.q.|| by Th19;
   hence thesis by A1;
end;
