reserve Y for RealNormSpace;

theorem FTh40:
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,Y holds R*I is RestFunc of Y)
 & for L being LinearOperator of REAL-NS 1,Y holds L*I is LinearFunc of Y
proof
   let I be Function of REAL,REAL-NS 1;
   assume A1: I=proj(1,1) qua Function";
A0:dom I = REAL by FUNCT_2:def 1;
   thus for R being RestFunc of REAL-NS 1,Y holds R*I is RestFunc of Y
   proof
    let R be RestFunc of REAL-NS 1,Y;
A2: R is total by NDIFF_1:def 5; then
    reconsider R0=R as Function of REAL 1,Y by Lm1;
    reconsider R1=R*I as PartFunc of REAL,Y;
A3: R0*I is Function of REAL,Y 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);
      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 A0,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.Y
    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;
       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 - 0.Y .|| by NDIFF_1:def 2;
       assume n0 <= m;
       then |.h.m - 0 .| < d by A18;
       hence ||. (h"(#)(R1/*h)).m - 0.Y .|| < r by A17,SEQ_1:5,A21;
      end;
     end;
     hence (h")(#)(R1/*h) is convergent;
     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,Y;
   reconsider L0=L as Function of REAL 1, Y by Lm1;
   reconsider L1=L0*I as PartFunc of REAL,Y;
   reconsider r = L1.jj as Point of Y by FUNCT_2:5;
A22: dom(L0*I) = REAL & dom L1 = REAL by 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;
    L1.p = L0.(I.(p*1)) by A0,XREAL_0:def 1,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 XREAL_0:def 1,A22,PARTFUN1:def 6;
    hence thesis by A22,FUNCT_1:12;
   end;
   hence thesis by NDIFF_3:def 2;
end;
