reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM100:
  for X,Y,Z be RealNormSpace,
      u,v be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
      w be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z)
  holds w*(u-v) = w*u-w*v & w*(u+v) = w*u+w*v
  proof
    let X,Y,Z be RealNormSpace,
        u,v be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
        w be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z);
    A2: modetrans(u,X,Y)=u by LOPBAN_1:def 11;
    for x be Point of X holds (w*(u-v)).x =(w*u).x-(w*v).x
    proof
      let x be Point of X;
      A6: modetrans(w,Y,Z) is additive;
      thus (w*(u-v)).x
       = modetrans(w,Y,Z).(modetrans(u-v,X,Y).x) by FUNCT_2:15
      .= modetrans(w,Y,Z). ((u-v).x) by LOPBAN_1:def 11
      .= modetrans(w,Y,Z).(u.x-v.x) by LOPBAN_1:40
      .= modetrans(w,Y,Z).(u.x)+modetrans(w,Y,Z).(-v.x) by A6
      .= modetrans(w,Y,Z).(u.x)+modetrans(w,Y,Z).((-1)*v.x) by RLVECT_1:16
      .= modetrans(w,Y,Z).(u.x)+(-1) * modetrans(w,Y,Z).(v.x) by LOPBAN_1:def 5
      .= modetrans(w,Y,Z).(u.x) -modetrans(w,Y,Z).(v.x) by RLVECT_1:16
      .= modetrans(w,Y,Z).(modetrans(u,X,Y).x)
         -modetrans(w,Y,Z).(modetrans(v,X,Y).x) by A2,LOPBAN_1:def 11
      .= (modetrans(w,Y,Z)*modetrans(u,X,Y)).x
         -modetrans(w,Y,Z).(modetrans(v,X,Y).x) by FUNCT_2:15
      .= (w*u).x- (w*v).x by FUNCT_2:15;
    end;
    hence w*(u-v) = w*u-w*v by LOPBAN_1:40;
    for x be Point of X holds (w*(u+v)).x = (w*u).x+(w*v).x
    proof
      let x be Point of X;
      A7: modetrans(w,Y,Z) is additive;
      thus (w*(u+v)).x
       = modetrans(w,Y,Z).(modetrans(u+v,X,Y).x) by FUNCT_2:15
      .= modetrans(w,Y,Z). ((u+v).x) by LOPBAN_1:def 11
      .= modetrans(w,Y,Z).(u.x+v.x) by LOPBAN_1:35
      .= modetrans(w,Y,Z).(u.x)+modetrans(w,Y,Z).(v.x) by A7
      .= modetrans(w,Y,Z).(modetrans(u,X,Y).x)
        +modetrans(w,Y,Z).(modetrans(v,X,Y).x) by A2,LOPBAN_1:def 11
      .= (modetrans(w,Y,Z)*modetrans(u,X,Y)).x
        +modetrans(w,Y,Z).(modetrans(v,X,Y).x) by FUNCT_2:15
      .= (w*u).x + (w*v).x by FUNCT_2:15;
    end;
    hence w*(u+v) = w*u+w*v by LOPBAN_1:35;
  end;
