reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM200:
  for X,Y,Z be RealNormSpace,
      w be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
      u,v be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z)
  holds (u-v)*w = u*w-v*w & (u+v)*w = u*w+v*w
  proof
    let X,Y,Z be RealNormSpace,
        w be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
      u,v be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z);
    A1: modetrans(w,X,Y)=w by LOPBAN_1:def 11;
    A2: modetrans(u,Y,Z)=u by LOPBAN_1:def 11;
    for x be Point of X holds ((u-v)*w).x =(u*w).x-(v*w).x
    proof
      let x be Point of X;
      thus ((u-v)*w).x
       = modetrans((u-v),Y,Z).(modetrans(w,X,Y).x) by FUNCT_2:15
      .= modetrans((u-v),Y,Z). (w.x) by LOPBAN_1:def 11
      .= (u-v).(w.x) by LOPBAN_1:def 11
      .= u.(w.x)-v.(w.x) by LOPBAN_1:40
      .= modetrans(u,Y,Z).(modetrans(w,X,Y).x)
        -modetrans(v,Y,Z).(modetrans(w,X,Y).x) by A1,A2,LOPBAN_1:def 11
      .= (modetrans(u,Y,Z)*modetrans(w,X,Y)).x
        -modetrans(v,Y,Z).(modetrans(w,X,Y).x) by FUNCT_2:15
      .= (u*w).x- (v*w).x by FUNCT_2:15;
    end;
    hence (u-v)*w = u*w-v*w by LOPBAN_1:40;
    for x be Point of X holds ((u+v)*w).x = (u*w).x+(v*w).x
    proof
      let x be Point of X;
      thus ((u+v)*w).x
       = modetrans((u+v),Y,Z).(modetrans(w,X,Y).x) by FUNCT_2:15
      .= modetrans((u+v),Y,Z). (w.x) by LOPBAN_1:def 11
      .= (u+v).(w.x) by LOPBAN_1:def 11
      .= u.(w.x)+v.(w.x) by LOPBAN_1:35
      .= modetrans(u,Y,Z).(modetrans(w,X,Y).x)
        +modetrans(v,Y,Z).(modetrans(w,X,Y).x) by A1,A2,LOPBAN_1:def 11
      .= (modetrans(u,Y,Z)*modetrans(w,X,Y)).x
        +modetrans(v,Y,Z).(modetrans(w,X,Y).x) by FUNCT_2:15
      .= (u*w).x+ (v*w).x by FUNCT_2:15;
    end;
    hence (u+v)*w = u*w+v*w by LOPBAN_1:35;
  end;
