reserve X,Y,Z for non trivial RealBanachSpace;

theorem
  for X be non trivial RealBanachSpace,
      v1,v2 be Lipschitzian LinearOperator of X,X,
      w1,w2 be Point of R_Normed_Algebra_of_BoundedLinearOperators X,
      a be Real
  st v1 = w1 & v2 = w2
  holds
     v1 * v2 = w1 * w2
   & v1 + v2 = w1 + w2
   & a (#) v1 =a * w1
  proof
    let X be non trivial RealBanachSpace,
        v1,v2 be Lipschitzian LinearOperator of X,X,
        w1,w2 be Point of R_Normed_Algebra_of_BoundedLinearOperators X,
        a be Real;
    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);
    v1 = modetrans(v1,X,X) & v2 = modetrans(v2,X,X) by LOPBAN_1:29;
    hence v1 * v2 = zw1 * zw2 by A1
     .= w1 * w2 by LOPBAN_2:def 4;
    reconsider zw1 = w1,zw2 = w2 as
      Point of R_NormSpace_of_BoundedLinearOperators(X,X);
    reconsider zw12 = zw1 + zw2 as
      Point of R_NormSpace_of_BoundedLinearOperators(X,X);
    zw12 is Lipschitzian LinearOperator of X,X by LOPBAN_1:def 9; then
    A4: dom zw12 = the carrier of X by FUNCT_2:def 1;
    A5: dom (v1 + v2)
     = dom v1 /\ dom v2 by VFUNCT_1:def 1
    .= (the carrier of X) /\ dom v2 by FUNCT_2:def 1
    .= (the carrier of X) /\ the carrier of X by FUNCT_2:def 1
    .= the carrier of X;
    for s being object st s in dom (v1 + v2) holds (v1 + v2).s = zw12.s
    proof
      let s be object;
      assume
      A6: s in dom (v1+v2); then
      reconsider d = s as Point of X;
      thus (v1 + v2).s = (v1 + v2)/.d by A5,PARTFUN1:def 6
        .= v1/.d + v2/.d by A6,VFUNCT_1:def 1
        .= zw12.s by A1,LOPBAN_1:35;
    end;
    hence v1 + v2 = w1 + w2 by A4,A5,FUNCT_1:2;
    reconsider zw12 = a * zw1 as
      Point of R_NormSpace_of_BoundedLinearOperators(X,X);
    zw12 is Lipschitzian LinearOperator of X,X by LOPBAN_1:def 9; then
    A8: dom zw12 = the carrier of X by FUNCT_2:def 1;
    A9: dom (a(#)v1)
      = dom v1 by VFUNCT_1:def 4
     .= the carrier of X by FUNCT_2:def 1;
    for s being object st s in dom (a(#)v1) holds (a(#)v1) . s = zw12 . s
    proof
      let s be object;
      assume
      A10: s in dom (a(#)v1); then
      reconsider d = s as Point of X;
      thus (a(#)v1) . s = (a(#)v1)/.d by A9,PARTFUN1:def 6
      .= a * v1/.d by A10,VFUNCT_1:def 4
      .= zw12.s by A1,LOPBAN_1:36;
    end;
    hence a(#)v1 = a * w1 by A8,A9,FUNCT_1:2;
  end;
