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

theorem Th1:
for S,T be RealNormSpace,
    f be Point of R_NormSpace_of_BoundedLinearOperators(S,T),
    r be Real st
  0 <= r &
  for x be Point of S st ||.x.|| <= 1 holds ||. f.x .|| <= r * ||.x.||
 holds ||.f.|| <= r
proof
   let S,T be RealNormSpace,
       f be Point of R_NormSpace_of_BoundedLinearOperators(S,T),
       r be Real;
   assume
A1: 0 <= r &
    for x be Point of S st ||.x.|| <= 1 holds ||. f.x .|| <= r * ||.x.||;
A2:now let x be Point of S;
    assume ||.x.|| <= 1; then
    ||. f.x .|| <= r * ||.x.|| & r * ||.x.|| <= r * 1 by A1,XREAL_1:64;
    hence ||. f.x .|| <= r by XXREAL_0:2;
   end;
   reconsider g = f as Lipschitzian LinearOperator of S,T by LOPBAN_1:def 9;
   set PreNormS = PreNorms(modetrans(f,S,T));
A3:for y be ExtReal st y in PreNorms(modetrans(f,S,T)) holds y <= r
   proof
    let y be ExtReal;
    assume y in PreNormS; then
    consider x be VECTOR of S such that
A4:  y = ||. modetrans(f,S,T).x .|| & ||.x.|| <= 1;
    y = ||. g.x .|| by A4,LOPBAN_1:29;
    hence thesis by A2,A4;
   end;
   set UBPreNormS = upper_bound PreNormS;
   set dif = UBPreNormS - r;
   now assume UBPreNormS > r; then
A5: dif > 0 by XREAL_1:50;
    r is UpperBound of PreNormS by A3,XXREAL_2:def 1; then
    PreNormS is bounded_above by XXREAL_2:def 10; then
    ex w being Real st
      w in PreNormS & UBPreNormS - dif < w by A5,SEQ_4:def 1;
    hence contradiction by A3;
   end; then
   upper_bound PreNorms(g) <= r by LOPBAN_1:def 11;
   hence thesis by LOPBAN_1:30;
end;
