reserve Y for RealNormSpace;

theorem FTh41:
for J be Function of REAL-NS 1,REAL st J = proj(1,1) holds
   (for R being RestFunc of Y holds R*J is RestFunc of REAL-NS 1,Y)
 & for L being LinearFunc of Y holds
     L*J is Lipschitzian LinearOperator of REAL-NS 1,Y
proof
   let J be Function of REAL-NS 1,REAL;
   assume A1: J=proj(1,1);
   thus for R being RestFunc of Y holds R*J is RestFunc of REAL-NS 1,Y
   proof
    let R be RestFunc of Y;
A2: R is total by NDIFF_3:def 1;
    reconsider R0=R as Function of REAL,Y by A2;
    reconsider R1 = R0*J as PartFunc of REAL-NS 1,Y;
    for h be (0.(REAL-NS 1))-convergent sequence of REAL-NS 1
      st h is non-zero
       holds ||.h.||"(#)(R1/*h) is convergent & lim(||.h.||"(#)(R1/*h)) = 0.Y
    proof
     let h be (0.(REAL-NS 1))-convergent sequence of REAL-NS 1;
     assume A3: h is non-zero;
A4:  lim h = 0.(REAL-NS 1) by NDIFF_1:def 4;
     deffunc F(Nat)=J.(h.$1);
     consider s be Real_Sequence such that
A5:   for n be Nat holds s.n = F(n) from SEQ_1:sch 1;
A6:  h is convergent by NDIFF_1:def 4;
A7:  now let p be Real;
      assume 0 < p;
      then consider m be Nat such that
A8:     for n be Nat st m <= n holds ||. h.n - 0.(REAL-NS 1).|| < p
           by A6,A4,NORMSP_1:def 7;
      take m;
      now let n be Nat;
       assume m <= n; then
A91:   ||. h.n - 0.(REAL-NS 1).|| < p by A8;
       s.n = J.(h.n) by A5;
       hence |.s.n-0 .| < p by A1,A91,PDIFF_1:4;
      end;
      hence for n be Nat st m <= n holds |.s.n-0 .|< p;
     end; then
     s is convergent by SEQ_2:def 6; then
A11: lim s = 0 by A7,SEQ_2:def 7;
     now let x be object;
      assume x in NAT;
      then reconsider n=x as Element of NAT;
A13:  ||. h.n .|| <> 0 by NORMSP_0:def 5,A3,NDIFF_1:6;
      s.n = J.(h.n) by A5;
      then |.s.n.| <> 0 by A1,A13,PDIFF_1:4;
      hence s.x <> 0 by COMPLEX1:47;
     end;
     then reconsider s as 0-convergent non-zero Real_Sequence
            by A7,SEQ_2:def 6,A11,FDIFF_1:def 1,SEQ_1:4;
     now
      reconsider f1=R1 as Function;
      let n be Element of NAT;
      rng h c= the carrier of REAL-NS 1; then
A14:  rng h c= dom R1 by FUNCT_2:def 1;
      (R/*s).n =R.(s.n) by NDIFF_3:def 1,FUNCT_2:115; then
A15:  (R/*s).n =R.(J.(h.n)) by A5;
      NAT = dom h by FUNCT_2:def 1; then
      R1.(h.n) =(f1*h).n by FUNCT_1:13; then
A17:  R1.(h.n) =(R1/*h).n by A14,FUNCT_2:def 11;
A18:  s.n = J.(h.n) by A5;
      ||. ||.h.||"(#)(R1/*h).|| .n
         = ||.(||.h.||"(#)(R1/*h)).n .|| by NORMSP_0:def 4
        .= ||.(||.h.||").n * (R1/*h).n .|| by NDIFF_1:def 2
        .= |.(||.h.||").n.| * ||.(R1/*h).n .|| by NORMSP_1:def 1
        .= |.(||.h.||.n)".| * ||.(R1/*h).n .|| by VALUED_1:10
        .= |.||.h.n.||".| * ||.(R1/*h).n .|| by NORMSP_0:def 4
        .= ||.h.n .||" *||.(R1/*h).n .|| by ABSVALUE:def 1
        .= (|.s.n.|)" *||.(R1/*h).n .|| by A1,A18,PDIFF_1:4
        .= (|.s.n.|)" *||.(R/*s).n .|| by A17,A15,FUNCT_2:15
        .= ((abs s).n)"*||.(R/*s).n .|| by SEQ_1:12
        .= ((abs s)".n)*||.(R/*s).n .|| by VALUED_1:10
        .=(|.s".|.n)*||.(R/*s).n .|| by SEQ_1:54
        .=|.s".n.|*||.(R/*s).n .|| by SEQ_1:12
        .= ||. (s".n)*(R/*s).n .|| by NORMSP_1:def 1
        .= ||. (s"(#)(R/*s)).n .|| by NDIFF_1:def 2
        .= ||. (s"(#)(R/*s)) .||.n by NORMSP_0:def 4;
      hence ||. ||.h.||"(#)(R1/*h) .|| .n = ||. (s"(#)(R/*s)) .||.n;
     end;
     then
A19: ||. ||.h.||"(#)(R1/*h) .|| = ||. s"(#)(R/*s) .|| by FUNCT_2:63;
     s"(#)(R/*s) is convergent & lim(s"(#)(R/*s))=0.Y by NDIFF_3:def 1; then
A22: lim ||. s"(#)(R/*s) .|| = ||. 0.Y .|| by LOPBAN_1:20;
A23: ||. s"(#)(R/*s) .|| is convergent by NDIFF_3:def 1,NORMSP_1:23;
A24: now let p be Real;
      assume 0 < p;
      then consider n0 be Nat such that
A25:    for m be Nat st n0 <= m holds |.||. ||.h.||"(#)(R1/*h).||.m - 0 .| < p
           by A19,A23,A22,SEQ_2:def 7;
      take n0;
      hereby let m be Nat;
       assume n0 <= m; then
       |. ||. ||.h.||"(#)(R1/*h).|| .m - 0 .| < p by A25; then
       |. ||.( ||.h.||"(#)(R1/*h)).m.|| .| < p by NORMSP_0:def 4;
       hence ||.(||.h.||"(#)(R1/*h)).m -0.(Y) .|| < p by ABSVALUE:def 1;
      end;
     end;
     then ||.h.||"(#)(R1/*h) is convergent;
     hence thesis by A24,NORMSP_1:def 7;
    end;
    hence thesis by NDIFF_1:def 5;
   end;
   let L be LinearFunc of Y;
   consider r be Point of Y such that
A27:for p be Real holds L/.p = p*r by NDIFF_3:def 2;
   reconsider L0 = L as Function of REAL,Y;
   set K = ||.r.||;
   reconsider L1 = L*J as Function of REAL-NS 1,Y;
A28: dom L1 = REAL 1 by Lm1,FUNCT_2:def 1;
A29:
   now let x,y be Point of REAL-NS 1;
    L1.(x+y) =L/.(J.(x+y)) by Lm1,A28,FUNCT_1:12;
    then L1.(x+y) = L/.((J.x+J.y)) by A1,PDIFF_1:4;
    then L1.(x+y) = (J.x+J.y)*r by A27;
    then L1.(x+y) = (J.x)*r+(J.y)*r by RLVECT_1:def 6;
    then L1.(x+y) = L/.(J.x)+(J.y)*r by A27;
    then
A30:L1.(x+y) = L/.(J.x)+L/.(J.y) by A27;
    L/.(J.x) = L1.x by Lm1,A28,FUNCT_1:12;
    hence L1.(x+y) =L1.x + L1.y by Lm1,A28,A30,FUNCT_1:12;
   end;
   now let x be Point of REAL-NS 1, a be Real;
    L1.(a*x) = L/.(J.(a*x)) by Lm1,A28,FUNCT_1:12;
    then L1.(a*x) = L/.(a*(J.x)) by A1,PDIFF_1:4;
    then L1.(a*x) = (a*(J.x))*r by A27;
    then
A31:L1.(a*x) = a*((J.x)*r) by RLVECT_1:def 7;
    L/.(J.x) = L1.x by Lm1,A28,FUNCT_1:12;
    hence L1.(a*x) =a*(L1.x) by A31,A27;
   end;
   then reconsider L1 as LinearOperator of REAL-NS 1,Y
         by A29,LOPBAN_1:def 5,VECTSP_1:def 20;
   now let x be Point of REAL-NS 1;
    ||. L1.x .|| =||. L/.(J.x) .|| by Lm1,A28,FUNCT_1:12;
    then ||. L1.x .|| =||. (J.x)*r .|| by A27;
    then ||. L1.x .|| =||.r.||*|.J.x .| by NORMSP_1:def 1;
    hence ||. L1.x .|| <= K* ||.x.|| by A1,PDIFF_1:4;
   end;
   hence thesis by LOPBAN_1:def 8;
end;
