
theorem Th6:
  for I be Function of REAL,REAL 1, J be Function of REAL 1,REAL st
I=proj(1,1) qua Function" & J=proj(1,1) holds
(for R being RestFunc holds I*R*J is
  RestFunc of REAL-NS 1,REAL-NS 1) &
  for L being LinearFunc holds I*L*J is Lipschitzian
  LinearOperator of REAL-NS 1,REAL-NS 1
proof
  let I be Function of REAL,REAL 1, J be Function of REAL 1,REAL;
  assume that
A1: I=proj(1,1) qua Function" and
A2: J=proj(1,1);
  thus for R being RestFunc holds I*R*J is RestFunc of REAL-NS 1,REAL-NS 1
  proof
    let R be RestFunc;
    R is total by FDIFF_1:def 2;
    then reconsider R0=R as Function of REAL,REAL;
A3: the carrier of REAL-NS 1 = REAL 1 by REAL_NS1:def 4;
    then reconsider R1 = I*R0*J as PartFunc of REAL-NS 1,REAL-NS 1;
    for h be (0.(REAL-NS 1))-convergent non-zero sequence of REAL-NS 1
     holds ||.h.||"(#)(R1/*h
    ) is convergent & lim(||.h.||"(#)(R1/*h)) = 0.(REAL-NS 1)
    proof
      let h be (0.(REAL-NS 1))-convergent non-zero sequence of REAL-NS 1;
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;
         reconsider m as Nat;
        take m;
        now
          let n be Nat;
          assume
A9:          m <= n;
           reconsider nn=n as Nat;
          ||. h.nn - 0.(REAL-NS 1).|| < p by A8,A9;
          then
A10:      ||. h.nn .|| < p by RLVECT_1:13;
          s.n = J.(h.n) by A5;
          hence |.s.n-0 .| < p by A2,A10,Th4;
        end;
        hence for n be Nat st m <= n holds |.s.n-0 .|< p;
      end;
      then
A11:  s is convergent by SEQ_2:def 6;
      then
A12:  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:    0 <= |.s.n.| by COMPLEX1:46;
        h.n <> 0.(REAL-NS 1) by NDIFF_1:6;
        then
A14:    ||. h.n .|| <> 0 by NORMSP_0:def 5;
        s.n = J.(h.n) by A5;
        then |.s.x.| <> 0 by A2,A14,Th4;
        hence s.x <> 0 by A13,COMPLEX1:47;
      end;
      then s is non-zero by SEQ_1:4;
      then reconsider s as 0-convergent non-zero Real_Sequence by A11,A12,
FDIFF_1:def 1;
A15:  J*I =id REAL by A1,A2,Lm1,FUNCT_1:39;
      now
        reconsider f1=R1 as Function;
        let n be Element of NAT;
A16:    rng h c= the carrier of REAL-NS 1;
        h.n in the carrier of (REAL-NS 1);
        then
A17:    h.n in (REAL 1) by REAL_NS1:def 4;
A18:     (R0*J).(h.n) in REAL by XREAL_0:def 1;
        R1 is total by A3;
        then (R/*s).n =R.(s.n) by FUNCT_2:115;
        then (R/*s).n =R.(J.(h.n)) by A5;
        then (R/*s).n =(J*I).(R0.(J.(h.n))) by A15;
        then (R/*s).n =(J*I).((R0*J).(h.n)) by A17,FUNCT_2:15;
        then (R/*s).n =J.(I.((R0*J).(h.n))) by FUNCT_2:15,A18;
        then
A19:    (R/*s).n =J.((I*(R0*J)).(h.n)) by A17,FUNCT_2:15;
        NAT = dom h by FUNCT_2:def 1;
        then
A20:    R1.(h.n) =(f1*h).n by FUNCT_1:13;
        dom R1 = REAL 1 by FUNCT_2:def 1;
        then rng h c= dom R1 by A16,REAL_NS1:def 4;
        then R1.(h.n) =(R1/*h).n by A20,FUNCT_2:def 11;
        then
A21:    (R/*s).n =J.((R1/*h).n) by A19,RELAT_1:36;
A22:    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 A2,A22,Th4
          .= (|.s.n.|)" *|.(R/*s).n.| by A2,A21,Th4
          .= (|.s.n.|)" *(|.R/*s.|).n by SEQ_1:12
          .= ((abs s).n)" *(|.R/*s.|).n by SEQ_1:12
          .= ((abs s)").n *(|.R/*s.|).n by VALUED_1:10
          .= ((abs s)"(#)|.R/*s.|).n by SEQ_1:8
          .= (|.s".|(#)abs(R/*s)).n by SEQ_1:54;
        hence ||. ||.h.||"(#)(R1/*h) .|| .n = (|.s"(#)(R/*s).|).n by SEQ_1:52;
      end;
      then
A23:  ||. ||.h.||"(#)(R1/*h) .|| = |.s"(#)(R/*s).| by FUNCT_2:63;
A24:  lim(s"(#)(R/*s))=0 by FDIFF_1:def 2;
A25:  s"(#)(R/*s) is convergent by FDIFF_1:def 2;
      then lim |.s"(#)(R/*s).| = |.lim(s"(#)(R/*s)).| by SEQ_4:14;
      then
A26:  lim |.s"(#)(R/*s).| =0 by A24,ABSVALUE:2;
A27:  abs(s"(#)(R/*s)) is convergent by A25,SEQ_4:13;
A28:  now
        let p be Real;
        assume 0 < p;
        then consider m be Nat such that
A29:    for n be Nat st m <= n holds |.||. ||.h.||"(#)(
        R1/*h).||.n - 0 .| < p by A23,A27,A26,SEQ_2:def 7;
         reconsider m as Nat;
        take m;
          let n be Nat;
          assume m <= n;
          then |.||. ||.h.||"(#)(R1/*h).|| .n - 0 .| < p by A29;
          then
A30:      |. ||.(||.h.||"(#)(R1/*h)).n.||.| < p by NORMSP_0:def 4;
          ||.(||.h.||"(#)(R1/*h)).n.|| < p by A30,ABSVALUE:def 1;
          hence ||.(||.h.||"(#)(R1/*h)).n -0.(REAL-NS 1).|| < p by RLVECT_1:13;
      end;
      then ||.h.||"(#)(R1/*h) is convergent by NORMSP_1:def 6;
      hence thesis by A28,NORMSP_1:def 7;
    end;
    hence thesis by A3,NDIFF_1:def 10;
  end;
  thus for L being LinearFunc holds I*L*J is Lipschitzian
  LinearOperator of REAL-NS 1,REAL-NS 1
  proof
    let L be LinearFunc;
    consider r be Real such that
A31: for p be Real holds L.p = r * p by FDIFF_1:def 3;
    L is total by FDIFF_1:def 3;
    then reconsider L0 = L as Function of REAL,REAL;
    reconsider r as Real;
    set K = |.r.|;
A32: the carrier of REAL-NS 1 = REAL 1 by REAL_NS1:def 4;
    I*L0*J is Function of REAL 1,REAL 1;
    then reconsider L1 = I*L*J as Function of REAL-NS 1,REAL-NS 1 by A32;
A33: dom L1 = REAL 1 by A32,FUNCT_2:def 1;
A34: dom L0 = REAL by FUNCT_2:def 1;
A35: now
      let x,y be VECTOR of REAL-NS 1;
A36:   J.x+J.y in REAL by XREAL_0:def 1;
A37:   J.x in REAL by XREAL_0:def 1;
      J.y in REAL by XREAL_0:def 1; then
      I.(L.(J.y)) = (I*L).(J.y) by A34,FUNCT_1:13;
      then
A38:  I.(L.(J.y)) = L1.y by A32,A33,FUNCT_1:12;
      L1.(x+y) =(I*L).(J.(x+y)) by A32,A33,FUNCT_1:12;
      then L1.(x+y) =(I*L).(J.x+J.y) by A2,Th4;
      then L1.(x+y) =I.(L.((J.x+J.y))) by A36,A34,FUNCT_1:13;
      then L1.(x+y) =I.(r*(J.x+J.y)) by A31;
      then L1.(x+y) =I.(r*J.x+r*J.y);
      then L1.(x+y) =I.(L.(J.x)+(r*J.y)) by A31;
      then
A39:  L1.(x+y) =I.(L.(J.x)+L.(J.y)) by A31;
      I.(L.(J.x)) = (I*L).(J.x) by A37,A34,FUNCT_1:13;
      then I.(L.(J.x)) = L1.x by A32,A33,FUNCT_1:12;
      hence L1.(x+y) =L1.x + L1.y by A1,A38,A39,Th3;
    end;
    now
      let x be VECTOR of REAL-NS 1, a be Real;
      reconsider aa=a as Real;
A40:   J.x in REAL by XREAL_0:def 1;
A41:   aa*J.x in REAL by XREAL_0:def 1;
      L1.(a*x) =(I*L).(J.(a*x)) by A32,A33,FUNCT_1:12;
      then L1.(a*x) =(I*L).(a*J.x) by A2,Th4;
      then L1.(aa*x) =I.(L.((aa*J.x))) by A41,A34,FUNCT_1:13;
      then L1.(a*x) =I.(r*(a*J.x)) by A31;
      then L1.(a*x) =I.(a*(r*J.x));
      then
A42:  L1.(a*x) =I.(a*L.(J.x)) by A31;
      I.(L.(J.x)) = (I*L).(J.x) by A40,A34,FUNCT_1:13;
      then I.(L.(J.x)) = L1.x by A32,A33,FUNCT_1:12;
      hence L1.(a*x) =a*(L1.x) by A1,A42,Th3;
    end;
    then reconsider L1 as LinearOperator of REAL-NS 1,REAL-NS 1 by A35,
VECTSP_1:def 20,LOPBAN_1:def 5;
A43: now
      let x be VECTOR of REAL-NS 1;
      J.x in REAL by XREAL_0:def 1; then
      I.(L.(J.x)) =(I*L).(J.x) by A34,FUNCT_1:13;
      then I.(L.(J.x)) =L1.x by A32,A33,FUNCT_1:12;
      then ||. L1.x .|| =|.L.(J.x).| by A1,Th3;
      then ||. L1.x .|| =|.r*J.x.| by A31;
      then ||. L1.x .|| =|.r.|*|.J.x.| by COMPLEX1:65;
      hence ||. L1.x .|| <= K* ||.x.|| by A2,Th4;
    end;
    0 <= K by COMPLEX1:46;
    hence thesis by A43,LOPBAN_1:def 8;
  end;
end;
