reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve y,X for set;
reserve x,x0,x1,x2,g,g1,g2,r,r1,s,p,p1,p2 for Real;
reserve i,m,k for Element of NAT;
reserve n,k for non zero Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq,seq1 for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve g,g1,g2 for PartFunc of REAL,REAL-NS n;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;
reserve GR,R for RestFunc of REAL-NS n;
reserve DFG,L for LinearFunc of REAL-NS n;
reserve m for non zero Element of NAT;

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;
