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 Th36:
  for X, Y be RealNormSpace for f,h be Point of
R_NormSpace_of_BoundedLinearOperators(X,Y)
for a be Real holds h = a*f iff for
  x be VECTOR of X holds h.x = a * f.x
proof
  let X, Y be RealNormSpace;
  let f,h be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
  let a be Real;
  reconsider f1=f as VECTOR of R_VectorSpace_of_BoundedLinearOperators(X,Y);
  reconsider h1=h as VECTOR of R_VectorSpace_of_BoundedLinearOperators(X,Y);
  h=a*f iff h1=a*f1;
  hence thesis by Th25;
end;
