reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem
  for X,Y,Z be RealNormSpace
  for f be Element of R_NormSpace_of_BoundedLinearOperators(X,Y),
    g,h be Element of R_NormSpace_of_BoundedLinearOperators(Y,Z)
  holds (g + h) * f = g*f + h*f
  proof
    let X,Y,Z be RealNormSpace;
    let f be Element of R_NormSpace_of_BoundedLinearOperators(X,Y),
      g,h be Element of R_NormSpace_of_BoundedLinearOperators(Y,Z);
    set BLOPXY = R_NormSpace_of_BoundedLinearOperators(X,Y);
    set BLOPXZ = R_NormSpace_of_BoundedLinearOperators(X,Z);
    set BLOPYZ = R_NormSpace_of_BoundedLinearOperators(Y,Z);
    set mf = modetrans(f,X,Y);
    set mg = modetrans(g,Y,Z);
    set mh = modetrans(h,Y,Z);
    set mgh= modetrans(g+h, Y,Z);
    A1: mh*mf is Lipschitzian LinearOperator of X,Z by LOPBAN_2:2; then
    reconsider hf = mh*mf as VECTOR of BLOPXZ by LOPBAN_1:def 9;
    A2: mg*mf is Lipschitzian LinearOperator of X,Z by LOPBAN_2:2; then
    reconsider gf=mg*mf as VECTOR of BLOPXZ by LOPBAN_1:def 9;
    A3: mgh*mf is Lipschitzian LinearOperator of X,Z by LOPBAN_2:2; then
    reconsider k = mgh*mf as VECTOR of BLOPXZ by LOPBAN_1:def 9;
    reconsider hh = h as VECTOR of BLOPYZ;
    reconsider gg = g as VECTOR of BLOPYZ;
    A4: gg = mg & hh = mh by LOPBAN_1:def 11;
    for x be VECTOR of X holds (mgh*mf).x=(mg*mf).x + (mh*mf).x
    proof
      let x be VECTOR of X;
      g+h = gg+hh & modetrans(g+h, Y,Z) = g+h by LOPBAN_1:def 11; then
      A5: mgh.(mf.x) = mg.(mf.x) + mh.(mf.x) by A4,LOPBAN_1:35;
      thus (mgh*mf).x = mgh.(mf.x) by A3,LPB2Th4
        .= (mg*mf).x + mh.(mf.x) by A2,A5,LPB2Th4
        .= (mg*mf).x + (mh*mf).x by A1,LPB2Th4;
    end;
    hence thesis by LOPBAN_1:35;
  end;
