reserve p for Real;
reserve S,T for RealNormSpace;
reserve x0 for Point of S;
reserve f for PartFunc of S,T;
reserve c for constant sequence of S;
reserve R for RestFunc of S,T;
reserve U for RealNormSpace;

theorem Th12:
  for R1 be RestFunc of S,T st R1/.0.S=0.T for R2 be RestFunc of T,U st R2
  /.0.T=0.U for L1 be Lipschitzian LinearOperator of S,T
  for L2 be Lipschitzian
  LinearOperator of T,U holds L2*R1+ R2*(L1+R1) is RestFunc of S,U
proof
  let R1 be RestFunc of S,T such that
A1: R1/.0.S=0.T;
  let R2 be RestFunc of T,U such that
A2: R2/.0.T=0.U;
  let L1 be Lipschitzian LinearOperator of S,T;
  let L2 be Lipschitzian LinearOperator of T,U;
A3: L2*R1 is RestFunc of S,U by Th9;
  R2*(L1+R1) is RestFunc of S,U by A1,A2,Th11;
  hence thesis by A3,NDIFF_1:28;
end;
