 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;

theorem Th18:
  for R be Function of REAL,REAL holds R is RestFunc-like iff
  for r be Real st r > 0
   ex d be Real st d > 0 & for z be Real
     st z <> 0 & |.z.| < d holds (|.R.z.|/ |.z.|) < r
proof
  let R be Function of REAL,REAL;
A1:now assume
A2: R is RestFunc-like;
    assume not
    (for r be Real st r > 0
    ex d be Real st d > 0 & for z be Real
    st z <> 0 & |.z.| < d holds ( |.R.z.|/ |.z.| ) < r); then
    consider r be Real such that
A3:  r > 0 and
A4:  for d be Real st d > 0 holds
    ex z be Real st z <> 0 & |.z.| < d
    & not ( |.R.z.|/|.z.|) < r;
    defpred P[Nat,Element of REAL]
     means $2 <> 0 & |.$2.| < 1/($1+1) & not |.R.$2.|/|.$2.| < r;

A5: now let n be Element of NAT;
     consider z be Real such that
A6:    z <> 0 & |.z.| < 1/(n + 1)
    & not ( |.R.z.|/|.z.|) < r by A4;
     reconsider z as Element of REAL by XREAL_0:def 1;
     take z;
     thus P[n,z] by A6;
    end;
    consider s be Real_Sequence such that
A7:  for n being Element of NAT holds P[n,s.n] from FUNCT_2:sch 3(A5);
A8:  for n being Nat holds P[n,s.n]
      proof let n be Nat;
        n in NAT by ORDINAL1:def 12;
       hence thesis by A7;
      end;
A9: now let p be Real;
     assume
A10:   0<p;
     consider n be Nat such that
A11:  p"<n by SEQ_4:3;
     p" + (0 qua Real) < n + 1 by A11,XREAL_1:8; then
A12: 1/(n+1) < 1/p" by A10,XREAL_1:76;
     take n;
     let m be Nat;
     assume n<=m; then
     n + 1 <= m + 1 by XREAL_1:6; then
     1/(m+1) <= 1/(n+1) by XREAL_1:118; then
     |.s.m-0 .| < 1/(n+1) by A8,XXREAL_0:2;
     hence |.s.m-0 .| < p by A12,XXREAL_0:2;
    end; then
    s is convergent by SEQ_2:def 6; then
    lim s = 0 by A9,SEQ_2:def 7; then
    reconsider s as 0-convergent non-zero Real_Sequence
    by A9,A8,SEQ_1:5,SEQ_2:def 6,FDIFF_1:def 1;
    (s")(#)(R/*s) is convergent & lim ((s")(#)(R/*s)) = 0
      by A2,FDIFF_1:def 2; then
    consider n be Nat such that
A15: for m be Nat st n <=m holds |.((s")(#)(R/*s)).m- 0 .| < r
      by A3,SEQ_2:def 7;
A16:   n in NAT by ORDINAL1:def 12;
A18: |.(s.n)"*(R.(s.n)).| = |.(s.n)".| * |.R.(s.n).| by COMPLEX1:65
      .= |.R.(s.n).|/|.s.n.| by COMPLEX1:66;
    |. ((s")(#)(R/*s)).n- 0 .|
      = |. (s".n)*((R/*s).n) .| by SEQ_1:8
      .= |. (s.n)"*((R/*s).n) .| by VALUED_1:10
      .= |. (s.n)"*(R.(s.n)) .| by FUNCT_2:115,A16;
    hence
     for r be Real st r > 0
     ex d be Real st d > 0 & for z be Real st
      z <> 0 & |.z.| < d holds ( |.R.z.|/|.z.| ) < r by A8,A15,A18;
   end;
   now assume
A19: for r be Real st r > 0
    ex d be Real st d > 0 & for z be Real
    st z <> 0 & |.z.| < d holds ( |.R.z.|/|.z.| ) < r;
    now let s be 0-convergent non-zero Real_Sequence;
A20: s is convergent & lim s = 0;
A21: now let r be Real;
      assume A22: r > 0;
      consider d be Real such that
A23:   d > 0 and
A24:   for z be Real st z <> 0 & |.z.| < d
     holds |.R.z.|/|.z.| < r by A22,A19;
      consider n be Nat such that
A25:   for m be Nat st n <= m holds |.s.m-0 .| < d
         by A20,A23,SEQ_2:def 7;
      take n;
      hereby let m be Nat;
A26:   m in NAT by ORDINAL1:def 12;
       assume n <=m; then
A27:   |.s.m-0 .| < d by A25;
       |.R.(s.m).| / |.s.m.|
         = |.(s.m)".| * |.R.(s.m).| by COMPLEX1:66
        .= |.(s.m)" * R.(s.m).| by COMPLEX1:65
        .= |.(s.m)" * (R/*s).m.| by FUNCT_2:115,A26
        .= |.s".m * (R/*s).m .| by VALUED_1:10
        .= |.(s"(#)(R/*s)).m - 0 .| by SEQ_1:8;
       hence |. ((s")(#)(R/*s)).m- 0 .| < r by A24,A27,SEQ_1:5;
      end;
     end;
     hence (s")(#)(R/*s) is convergent by SEQ_2:def 6;
     hence lim((s")(#)(R/*s)) = 0 by A21,SEQ_2:def 7;
    end;
    hence R is RestFunc-like by FDIFF_1:def 2;
   end;
   hence thesis by A1;
end;
