
theorem
  for V,W be RealLinearSpace, L be LinearOperator of V,W
  holds L = InducedBi(V,W,L) * InducedSur(V,Ker L)
  proof
    let V,W be RealLinearSpace, L be LinearOperator of V,W;
    now
      let v be VECTOR of V;
      A1: dom InducedSur(V,Ker L) = the carrier of V by FUNCT_2:def 1;
      reconsider z = v + Ker L as Point of VectQuot(V,Ker L) by LMQ07;
      thus (InducedBi(V,W,L) * InducedSur(V,Ker L)).v
         = (InducedBi(V,W,L)).((InducedSur(V,Ker L)).v) by A1,FUNCT_1:13
        .= (InducedBi(V,W,L)).z by defDQuot
        .= L.v by defQuotR;
    end;
    hence thesis;
  end;
