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

theorem Th18:
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*>
 & (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.||)
proof
   let f be Point of
          R_NormSpace_of_BoundedLinearOperators(REAL-NS 1,REAL-NS j);
   reconsider One = <*jj*> as Element of REAL 1 by FINSEQ_2:98;
   reconsider L = f as Lipschitzian LinearOperator of REAL-NS 1,REAL-NS j
      by LOPBAN_1:def 9;
   the carrier of REAL-NS 1 = REAL 1 by REAL_NS1:def 4; then
   dom L = REAL 1 by FUNCT_2:def 1; then
   reconsider p = f.<*jj*> as Point of REAL-NS j by FINSEQ_2:98,PARTFUN1:4;
   reconsider OneNS = One as VECTOR of REAL-NS 1 by REAL_NS1:def 4;
A1:now let r be Real, x be Point of REAL-NS 1;
    assume x = <*r*>; then
A2: f.x = L.<*r*>;
    <*r*> = <* r*1 *>
         .= r*<*1*> by RVSUM_1:47
         .= r*OneNS by REAL_NS1:3;
    hence f.x = r*p by A2,LOPBAN_1:def 5;
   end;
   now let x be Point of REAL-NS 1;
A3: the carrier of REAL-NS 1 = REAL 1 by REAL_NS1:def 4; then
    reconsider x0=x as FinSequence of REAL by FINSEQ_2:def 3;
    consider r be Element of REAL such that
A4:  x0 = <*r*> by A3,FINSEQ_2:97;
    thus ||. f.x .|| = ||. r*p .|| by A1,A4
               .= |.r.| * ||.p.|| by NORMSP_1:def 1
               .= ||.p.|| * ||.x.|| by A4,PDIFF_8:2;
   end;
   hence thesis by A1;
end;
