theorem Th49:
  for R1 be RestFunc of REAL-NS n st R1/.0=0.(REAL-NS n)
  for R2 be RestFunc of REAL-NS n,REAL-NS m st R2/.0.(REAL-NS n)=0.(REAL-NS m)
  for L1 be LinearFunc of REAL-NS n
  for L2 be Lipschitzian LinearOperator of REAL-NS n,REAL-NS m holds
    L2*R1+ R2*(L1+R1) is RestFunc of REAL-NS m
proof
  let R1 be RestFunc of REAL-NS n such that
A1: R1/.0=0.(REAL-NS n);
  let R2 be RestFunc of REAL-NS n,REAL-NS m such that
A2: R2/.0.(REAL-NS n)=0.(REAL-NS m);
  let L1 be LinearFunc of REAL-NS n;
  let L2 be Lipschitzian LinearOperator of REAL-NS n,REAL-NS m;
A3: L2*R1 is RestFunc of REAL-NS m by Th47;
  R2*(L1+R1) is RestFunc of REAL-NS m by A1,A2,Th48;
  hence thesis by A3,NDIFF_3:7;
end;
