
theorem Th11:
  for X be RealNormSpace for f,g be Element of
  BoundedLinearOperators(X,X)
  for a,b be Real holds (a*b)*(f*g)=(a*f)*(b*g)
proof
  let X be RealNormSpace;
  let f,g be Element of BoundedLinearOperators(X,X);
  let a,b be Real;
  set BLOP=R_NormSpace_of_BoundedLinearOperators(X,X);
  set EXMULT=Mult_(BoundedLinearOperators(X,X),
  R_VectorSpace_of_LinearOperators(X,X));
  set mf=modetrans(f,X,X);
  set mg=modetrans(g,X,X);
  set maf=modetrans((a*f),X,X);
  set mbg=modetrans(b*g,X,X);
  EXMULT.(a*b,mf*mg)=maf*mbg
  proof
    reconsider k=(maf)*(mbg) as VECTOR of BLOP by LOPBAN_1:def 9;
    reconsider fg=mf*mg as VECTOR of BLOP by LOPBAN_1:def 9;
    reconsider ff = f, gg = g as VECTOR of BLOP;
A1: gg=mg by LOPBAN_1:def 11;
A2: ff=mf by LOPBAN_1:def 11;
    for x be VECTOR of X holds ( (maf)*(mbg)).x=(a*b)*(mf*mg).x
    proof
      let x be VECTOR of X;
      set y=b*mg.x;
      a*f=a*ff & modetrans(a*f, X,X) =a*f by LOPBAN_1:def 11;
      then
A3:   maf.y=a*mf.y by A2,LOPBAN_1:36;
      b*g=b*gg & modetrans(b*g, X,X) =b*g by LOPBAN_1:def 11;
      then
A4:   mbg.x=b*mg.x by A1,LOPBAN_1:36;
      thus (maf*mbg).x=maf.(mbg.x) by Th4
        .=a*(b*mf.(mg.x)) by A3,A4,LOPBAN_1:def 5
        .=(a*b)*mf.(mg.x) by RLVECT_1:def 7
        .=(a*b)*(mf*mg).x by Th4;
    end;
    then k=(a*b)*fg by LOPBAN_1:36;
    hence thesis;
  end;
  hence thesis;
end;
