reserve E, F, G,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 Th13:
  for X, Y, Z be RealNormSpace,
      v be Lipschitzian LinearOperator of X,Y,
      s be Lipschitzian LinearOperator of Y,Z,
      pv be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
      ps be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z)
    st v = pv & s=ps
  holds s*v = ps*pv
proof
  let X, Y, Z be RealNormSpace,
      v be Lipschitzian LinearOperator of X,Y,
      s be Lipschitzian LinearOperator of Y,Z,
      pv be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
      ps be Point of R_NormSpace_of_BoundedLinearOperators (Y,Z);
  assume A1: v = pv & s=ps; then
  modetrans(pv,X,Y) = v by LOPBAN_1:29;
  hence s*v = ps*pv by A1,LOPBAN_1:29;
end;
