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 Th48:
  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 L be LinearFunc of REAL-NS n holds R2*(L+R1) is RestFunc of REAL-NS m
proof
  set S=REAL-NS n;
  set T=REAL-NS m;
  let R1 be RestFunc of S;
  assume R1/.0=0.S;
  then consider d0 be Real such that
A1: 0 < d0 and
A2: for h be Real st |.h.| < d0 holds ||.R1/.h.|| <=1* |.h.| by Th46;
  let R2 be RestFunc of REAL-NS n,REAL-NS m such that
A3: R2/.0.S=0.T;
  let L be LinearFunc of S;
  consider r be Point of S such that
A4: for h be Real holds L/.h = h*r by NDIFF_3:def 2;
  set K=||.r.||;
  R2 is total by NDIFF_1:def 5;
  then dom R2 = the carrier of S by PARTFUN1:def 2;
  then
A5: rng (L+R1) c= dom R2;
  R1 is total by NDIFF_3:def 1;
  then
A6: L+R1 is total by VFUNCT_1:32;
  then
A7: dom(L+R1)=REAL by PARTFUN1:def 2;
A8: now
    let ee be Real such that
A9: ee > 0;
    set e=ee/2;
A10: e < ee by A9,XREAL_1:216;
    set e1=e/(1 + K);
    e > 0 by A9,XREAL_1:215;
    then 0/(1 + K) < e/(1 + K) by XREAL_1:74;
    then consider d be Real such that
A11: 0 < d and
A12: for z be Point of S st ||.z.|| < d holds ||.R2/.z.|| <= e1*||.z.||
      by A3,NDIFF_2:7;
    set d1=d/(1 + K);
    set dd1=min(d0,d1);
A13: dd1 <=d1 by XXREAL_0:17;
A14: dd1 <=d0 by XXREAL_0:17;
A15: now
      let hh be Real such that
A16:  hh <> 0 and
A17:  |.hh.| < dd1;
      |.hh.| < d0 by A14,A17,XXREAL_0:2;
      then
A18:  ||.R1/.hh.|| <=1* |.hh.| by A2;
      reconsider h = hh as Element of REAL by XREAL_0:def 1;
A19:  L/.h = h*r by A4;
      reconsider p0=In(0,REAL) as Element of REAL;
      ||. L/.h .|| - K * |.h.| + K * |.h.| <= p0 + K * |.h.|
               by A19,NORMSP_1:def 1; then
      ||.L/.h+R1/.h.|| <= ||.L/.h.|| + ||.R1/.h.|| & ||.L/.h.|| + ||.R1/.h
      .|| <= K * |.h.| + 1* |.h.| by A18,NORMSP_1:def 1,XREAL_1:7;
      then
A20:  ||.L/.h+R1/.h.|| <= ( K +1) * |.h.| by XXREAL_0:2;
      |.h.| < d1 by A13,A17,XXREAL_0:2;
      then (K +1) * |.h.| < (K +1) *d1 by XREAL_1:68;
      then ||.L/.h+R1/.h.|| < (K +1) * d1 by A20,XXREAL_0:2;
      then ||.L/.h+R1/.h.|| < d by XCMPLX_1:87;
      then
A21:  ||.R2/.(L/.h+R1/.h).|| <= e1*||.L/.h+R1/.h.|| by A12;
      e1*||.L/.h+R1/.h.|| <= e1* ((K +1) * |.h.|) by A9,A20,XREAL_1:64;
      then
A22:  ||.R2/.(L/.h+R1/.h).|| <= e1* ((K +1) * |.h.|) by A21,XXREAL_0:2;
A23:  R2/.(L/.h+R1/.h) = R2/.(L/.h+R1/.h) .=R2/.((L+R1)/.h)
          by A7,VFUNCT_1:def 1
        .=(R2*(L+R1))/.h by A7,A5,PARTFUN2:5;
A24:  |.h.| <> 0 by A16,COMPLEX1:45;
      then |.h.| > 0 by COMPLEX1:46;
      then
      |.h.|"* ||.(R2*(L+R1))/.h.|| <= |.h.|"* (e1* ( K +1) * |.h .|)
        by A23,A22,XREAL_1:64;
      then |.h.|"* ||.(R2*(L+R1))/.h.|| <= |.h.|*|.h.|"*e1* (K +1);
      then |.h.|"* ||.(R2*(L+R1))/.h.|| <= 1*e1* (K +1) by A24,XCMPLX_0:def 7;
      then |.h.|"* ||.(R2*(L+R1))/.h.|| <= e by XCMPLX_1:87;
      hence |.hh.|"* ||.(R2*(L+R1))/.hh.|| < ee by A10,XXREAL_0:2;
    end;
    0/(1 + K) < d/(1 + K) by A11,XREAL_1:74;
    then 0 < dd1 by A1,XXREAL_0:15;
    hence ex dd1 be Real st dd1 > 0 &
   for h be Real st h <> 0 & |.h.|
    < dd1 holds |.h.|"* ||.(R2*(L+R1))/.h.|| < ee by A15;
  end;
  dom (R2*(L+R1)) = dom(L+R1) by A5,RELAT_1:27
    .=REAL by A6,PARTFUN1:def 2;
  then R2*(L+R1) is total by PARTFUN1:def 2;
  hence thesis by A8,Th23;
end;
