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 Th31:
  for X, Y be RealNormSpace holds (the carrier of X) --> 0.Y = 0.
  R_NormSpace_of_BoundedLinearOperators(X,Y)
proof
  let X, Y be RealNormSpace;
  ((the carrier of X) --> 0.Y) =0.R_VectorSpace_of_BoundedLinearOperators(
  X,Y) by Th26
    .=0.R_NormSpace_of_BoundedLinearOperators(X,Y);
  hence thesis;
end;
