 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;

theorem Th5:
  for R1 be RestFunc of S st R1/.0=0.S
  for R2 be RestFunc of S,T st R2/.(0.S)=0.T
  for L1 be LinearFunc of S
  for L2 be Lipschitzian LinearOperator of S,T holds
    L2*R1 + R2*(L1+R1) is RestFunc of T
proof
   let R1 be RestFunc of S such that
A1:  R1/.0=0.S;
   let R2 be RestFunc of S,T such that
A2:  R2/.(0.S)=0.T;
   let L1 be LinearFunc of S;
   let L2 be Lipschitzian LinearOperator of S,T;
   L2*R1 is RestFunc of T & R2*(L1+R1) is RestFunc of T
   by A1,A2,Th4,Th3;
   hence thesis by NDIFF_3:7;
end;
