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

theorem Th30:
  for X, Y be RealNormSpace for f be Lipschitzian LinearOperator of X,Y
  holds BoundedLinearOperatorsNorm(X,Y).f = upper_bound PreNorms(f)
proof
  let X, Y be RealNormSpace;
  let f be Lipschitzian LinearOperator of X,Y;
  reconsider f9=f as set;
  f in BoundedLinearOperators(X,Y) by Def9;
  hence BoundedLinearOperatorsNorm(X,Y).f =
  upper_bound PreNorms(modetrans(f9,X,Y)) by Def13
    .= upper_bound PreNorms(f) by Th29;
end;
