reserve X,Y,Z for non trivial RealBanachSpace;

theorem
  for X be non trivial RealBanachSpace,
  v1,v2 be Point of R_NormSpace_of_BoundedLinearOperators(X,X),
  w1,w2 be Point of R_Normed_Algebra_of_BoundedLinearOperators X
  st v1 = w1 & v2 = w2
  holds v1 * v2 = w1 * w2
  proof
    let X be non trivial RealBanachSpace,
    v1,v2 be Point of R_NormSpace_of_BoundedLinearOperators(X,X),
    w1,w2 be Point of R_Normed_Algebra_of_BoundedLinearOperators X;
    set S = R_Normed_Algebra_of_BoundedLinearOperators X;
    assume
    A1: v1= w1 & v2= w2;
    reconsider zw1 = w1,zw2 = w2 as Element of BoundedLinearOperators (X,X);
    thus v1*v2 = zw1 * zw2 by A1
    .= w1 * w2 by LOPBAN_2:def 4;
  end;
