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;

theorem Th40:
  for I be Function of REAL,REAL-NS 1 st
  I=proj(1,1) qua Function" holds
  (for R being RestFunc of REAL-NS 1,REAL-NS n holds
      R*I is RestFunc of REAL-NS n) &
  for L being LinearOperator of REAL-NS 1,REAL-NS n holds
     L*I is LinearFunc of REAL-NS n
proof
  let I be Function of REAL,REAL-NS 1;
  assume A1: I=proj(1,1) qua Function";
  thus for R being RestFunc of REAL-NS 1,
    REAL-NS n holds R*I is RestFunc of REAL-NS n
  proof
    let R be RestFunc of REAL-NS 1,REAL-NS n;
A2: R is total by NDIFF_1:def 5;
    reconsider R0=R as Function of REAL 1,REAL n
    by A2,Lm1,REAL_NS1:def 4;
    reconsider R1=R*I as PartFunc of REAL,REAL-NS n;
A3: R0*I is Function of REAL,REAL n by Lm1;
    then
A4: dom R1 = REAL by FUNCT_2:def 1;
A5: for r be Real st r > 0
     ex d be Real st d > 0 & for z1 be Real st
      z1 <> 0 & |. z1 .| < d holds |. z1 .|"*||. R1/.z1 .|| < r
    proof
      let r be Real;
      assume r > 0;
      then consider d be Real such that
A6:   d > 0 and
A7:   for z be Point of REAL-NS 1 st z <> 0.(REAL-NS 1) & ||.z.|| < d
      holds ||.z.||"*||. R/.z .|| < r by A2,NDIFF_1:23;
      take d;
      for z1 be Real st z1 <> 0 & |. z1 .| < d holds
      |. z1 .|" * ||. R1/.z1 .|| < r
      proof
        let z1 be Real such that
A8:     z1 <> 0 and
A9:    |.z1.| < d;
        reconsider zz=z1 as Element of REAL by XREAL_0:def 1;
        reconsider z = I.zz as Point of REAL-NS 1;
        |.zz.| > 0 by A8,COMPLEX1:47;
        then ||.z.|| <> 0 by A1,Lm1,PDIFF_1:3;
        then
A10:    z <> 0.(REAL-NS 1);
A11:    dom I = REAL by FUNCT_2:def 1;
        R is total by NDIFF_1:def 5;
        then dom R = the carrier of REAL-NS 1 by PARTFUN1:def 2;
        then R/.z = R.(I.z1) by PARTFUN1:def 6;
        then R/.z = R1.zz by A11,FUNCT_1:13;
        then
A12:    ||. R/.z .|| =||. R1/.zz .|| by A4,PARTFUN1:def 6;
A13:    ||.z.||" = |.z1.|" by A1,Lm1,PDIFF_1:3;
        ||.z.|| < d by A1,A9,Lm1,PDIFF_1:3;
        hence thesis by A7,A10,A13,A12;
      end;
      hence thesis by A6;
    end;
    for h be 0-convergent non-zero Real_Sequence holds h"(#)(R1/*h) is
    convergent & lim(h"(#)(R1/*h)) = 0.(REAL-NS n)
    proof
      let h be 0-convergent non-zero Real_Sequence;
A14:  now
        let r be Real;
A15:    lim h = 0;
        assume r > 0;
        then consider d be Real such that
A16:    d > 0 and
A17:    for z1 be Real st z1 <> 0 & |. z1 .| < d holds
          |. z1 .|" * ||. R1/.z1 .|| < r by A5;
        reconsider d1 =d as Real;
        consider n0 be Nat such that
A18:    for m be Nat st n0 <= m holds |.h.m-0 .| < d1 by A16,A15,SEQ_2:def 7;
        take n0;
        hereby
          let m be Nat;
A19:  m in NAT by ORDINAL1:def 12;
A20:      h.m <> 0 by SEQ_1:5;
          rng h c= dom R1 by A4;
          then
A21:      |.h.m.|" * ||. R1/.(h.m) .|| = |.h.m.|" * ||. (R1/*h).m .||
             by A19,FUNCT_2:109
            .= ((abs h).m)" * ||. (R1/*h).m .|| by SEQ_1:12
            .= (abs h)".m * ||. (R1/*h).m .|| by VALUED_1:10
            .= |.h".|.m * ||. (R1/*h).m .|| by SEQ_1:54
            .= |.h".m.| * ||. (R1/*h).m .|| by SEQ_1:12
            .= ||. h".m * (R1/*h).m .|| by NORMSP_1:def 1
            .= ||. (h"(#)(R1/*h)).m .|| by NDIFF_1:def 2
            .= ||. (h"(#)(R1/*h)).m - 0.(REAL-NS n) .|| by RLVECT_1:13;
          assume n0 <= m;
          then |.h.m - 0 .| < d by A18;
          hence ||. (h"(#)(R1/*h)).m - 0.(REAL-NS n) .|| < r
                   by A17,A20,A21;
        end;
      end;
      hence (h")(#)(R1/*h) is convergent by NORMSP_1:def 6;
      hence thesis by A14,NORMSP_1:def 7;
    end;
    hence thesis by A3,NDIFF_3:def 1;
  end;
  let L be LinearOperator of REAL-NS 1,REAL-NS n;
  reconsider L0=L as Function of REAL 1, REAL n by Lm1,REAL_NS1:def 4;
  reconsider L1=L0*I as PartFunc of REAL,REAL-NS n by REAL_NS1:def 4;
    reconsider r = L1.jj as Point of (REAL-NS n) by FUNCT_2:5;
A22: dom(L0*I) = REAL by Lm1,FUNCT_2:def 1;
     for p be Real holds L1/.p = p*r
     proof
      reconsider 1p = I.jj as VECTOR of REAL-NS 1;
      let p be Real;
A23:    p in REAL by XREAL_0:def 1;
A24:    dom L1 = REAL by FUNCT_2:def 1;
      dom I = REAL by FUNCT_2:def 1;
      then L1.p = L0.(I.(p*1)) by A23,FUNCT_1:13;
      then L1.p = L.(p*1p) by A1,Lm1,PDIFF_1:3;
      then L1.p = p*(L/.1p) by LOPBAN_1:def 5;
      then L1/.p = p*(L/.1p) by A23,A24,PARTFUN1:def 6;
      hence thesis by A22,FUNCT_1:12;
    end;
    hence thesis by NDIFF_3:def 2;
end;
