reserve X for non empty set;
reserve Y for ComplexLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Complex;
reserve u,v,w for VECTOR of CLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th26:
  for X,Y be ComplexNormSpace, g be Lipschitzian LinearOperator of X,Y
  holds PreNorms(g) is bounded_above
proof
  let X,Y be ComplexNormSpace;
  let g be Lipschitzian LinearOperator of X,Y;
  PreNorms(g) is bounded_above
  proof
    consider K be Real such that
A1: 0 <= K and
A2: for x be VECTOR of X holds ||. g.x .|| <= K*||. x .|| by Def6;
    take K;
      let r be ExtReal;
      assume r in PreNorms(g);
      then consider t be VECTOR of X such that
A3:   r=||.g.t.|| and
A4:   ||.t.|| <= 1;
A5:   ||.g.t.|| <= K*||. t .|| by A2;
      K*||. t .|| <= K *1 by A1,A4,XREAL_1:64;
      hence r <=K by A3,A5,XXREAL_0:2;
  end;
  hence thesis;
end;
