reserve h,h1,h2 for 0-convergent non-zero Real_Sequence,
  c,c1 for constant Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x0,r,r0,r1,r2,g,g1,g2 for Real,
  n0,k,n,m for Element of NAT,
  a,b,d for Real_Sequence,
  x for set;

theorem Th4:
  (for h,c st rng c ={x0} & rng (h+c) c= dom f & (for n being Nat holds h.n>0
) holds h"(#)(f/*(h+c) - f/*c) is convergent) & {x0} c= dom f implies for h1,h2
,c st rng c ={x0} & rng (h1 + c)c= dom f & rng (h2 + c) c= dom f &
 (for n being Nat holds
h1.n >0) & (for n being Nat holds h2.n >0)
 holds lim (h1"(#)(f/*(h1+c) - f/*c)) = lim (h2
  "(#)(f/*(h2+c) - f/*c))
proof
  assume that
A1: for h,c st rng c ={x0} & rng (h+c) c= dom f & (for n being Nat holds h.n>0)
  holds h"(#)(f/*(h+c) - f/*c) is convergent and
A2: {x0} c= dom f;
  let h1,h2,c such that
A3: rng c ={x0} and
A4: rng (h1 + c) c= dom f and
A5: rng (h2 + c) c= dom f and
A6: for n being Nat holds h1.n >0 and
A7: for n being Nat holds h2.n >0;
  deffunc G(Element of NAT) = h2.$1;
  deffunc F(Element of NAT) = h1.$1;
  consider d such that
A8: for n holds d.(2*n)=F(n) & d.(2*n+1)=G(n) from SCHEME1:sch 2;
  now
    let n be Nat;
    consider m such that
A9: n = 2*m or n = 2*m+1 by SCHEME1:1;
    now
      per cases by A9;
      suppose
        n = 2*m;
        then d.n = h1.m by A8;
        hence d.n<>0 by SEQ_1:5;
      end;
      suppose
        n = 2*m+1;
        then d.n = h2.m by A8;
        hence d.n<>0 by SEQ_1:5;
      end;
    end;
    hence d.n<>0;
  end;
  then
A10: d is non-zero by SEQ_1:5;
A11: h2 is convergent & lim h2 =0;
  h1 is convergent & lim h1 =0;
  then d is convergent & lim d =0 by A8,A11,FDIFF_2:1;
  then reconsider d as 0-convergent non-zero Real_Sequence
    by A10,FDIFF_1:def 1;
  deffunc F(Nat) = 2*$1;
  consider a such that
A12: for n being Nat holds a.n = F(n) from SEQ_1:sch 1;
  deffunc G(Nat) = 2*$1+1;
  consider b such that
A13: for n being Nat holds b.n = G(n) from SEQ_1:sch 1;
   for n holds b.n = G(n) by A13;
  then reconsider b as increasing sequence of NAT by FDIFF_2:3;
A14: rng (d+c) c= dom f
  proof
    let x be object;
    assume x in rng (d+c);
    then consider n such that
A15: x = (d+c).n by FUNCT_2:113;
    consider m such that
A16: n=2*m or n=2*m+1 by SCHEME1:1;
    now
      per cases by A16;
      suppose
        n = 2*m;
        then x = d.(2*m) + c.(2*m) by A15,SEQ_1:7
          .= h1.m + c.(2*m) by A8
          .= h1.m + c.m by VALUED_0:23
          .= (h1+c).m by SEQ_1:7;
        then x in rng(h1+c) by VALUED_0:28;
        hence thesis by A4;
      end;
      suppose
        n = 2*m+1;
        then x = d.(2*m+1) + c.(2*m+1) by A15,SEQ_1:7
          .= h2.m + c.(2*m+1) by A8
          .= h2.m + c.m by VALUED_0:23
          .= (h2 + c).m by SEQ_1:7;
        then x in rng(h2+c) by VALUED_0:28;
        hence thesis by A5;
      end;
    end;
    hence thesis;
  end;
  now
    let n;
    thus ((d"(#)(f/*(d+c) - f/*c))*b).n = (d"(#) (f/*(d+c) - f/*c)).(b.n) by
FUNCT_2:15
      .= (d"(#)(f/*(d+c) - f/*c)).(2*n+1) by A13
      .= (d").(2*n+1) * (f/*(d+c) - f/*c).(2*n+1) by SEQ_1:8
      .= (d.(2*n+1))" * (f/*(d+c) - f/*c).(2*n+1) by VALUED_1:10
      .= (h2.n)" * (f/*(d+c) - f/*c).(2*n+1) by A8
      .= (h2").n * (f/*(d+c) - f/*c).(2*n+1) by VALUED_1:10
      .= (h2").n *((f/*(d+c)).(2*n+1) - (f/*c).(2*n+1)) by RFUNCT_2:1
      .= (h2").n *(f.((d+c).(2*n+1)) - (f/*c).(2*n+1)) by A14,FUNCT_2:108
      .= (h2").n *(f.((d+c).(2*n+1)) - f.(c.(2*n+1))) by A2,A3,FUNCT_2:108
      .= (h2").n *(f.(d.(2*n+1)+c.(2*n+1)) - f.(c.(2*n+1))) by SEQ_1:7
      .= (h2").n *(f.(h2.n+c.(2*n+1)) - f.(c.(2*n+1))) by A8
      .= (h2").n *(f.(h2.n+c.(2*n+1)) - f.(c.n)) by VALUED_0:23
      .= (h2").n *(f.(h2.n+c.n) - f.(c.n)) by VALUED_0:23
      .= (h2").n *(f.((h2+c).n) - f.(c.n)) by SEQ_1:7
      .= (h2").n *((f/*(h2+c)).n - f.(c.n)) by A5,FUNCT_2:108
      .= (h2").n *((f/*(h2+c)).n - (f/*c).n) by A2,A3,FUNCT_2:108
      .= (h2").n *(f/*(h2+c) - f/*c).n by RFUNCT_2:1
      .= (h2"(#)(f/*(h2+c) - f/*c)).n by SEQ_1:8;
  end;
  then
A17: h2"(#)(f/*(h2+c) - f/*c) is subsequence of d"(#)(f/*(d+c) - f/*c) by
FUNCT_2:63;
  for n being Nat holds d.n > 0
  proof
    let n be Nat;
    consider m such that
A18: n = 2*m or n = 2*m+1 by SCHEME1:1;
    now
      per cases by A18;
      suppose
        n = 2*m;
        then d.n = h1.m by A8;
        hence thesis by A6;
      end;
      suppose
        n = 2*m+1;
        then d.n = h2.m by A8;
        hence thesis by A7;
      end;
    end;
    hence thesis;
  end;
  then
A19: d"(#)(f/*(d+c) - f/*c) is convergent by A1,A3,A14;
  for n holds a.n = F(n) by A12;
  then reconsider a as increasing sequence of NAT by FDIFF_2:2;
  now
    let n;
    thus ((d"(#)(f/*(d+c) - f/*c))*a).n = (d"(#) (f/*(d+c) - f/*c)).(a.n) by
FUNCT_2:15
      .= (d"(#)(f/*(d+c) - f/*c)).(2*n) by A12
      .= (d").(2*n) * (f/*(d+c) - f/*c).(2*n) by SEQ_1:8
      .= (d.(2*n))" * (f/*(d+c) - f/*c).(2*n) by VALUED_1:10
      .= (h1.n)" * (f/*(d+c) - f/*c).(2*n) by A8
      .= (h1").n * (f/*(d+c) - f/*c).(2*n) by VALUED_1:10
      .= (h1").n *((f/*(d+c)).(2*n) - (f/*c).(2*n)) by RFUNCT_2:1
      .= (h1").n *(f.((d+c).(2*n)) - (f/*c).(2*n)) by A14,FUNCT_2:108
      .= (h1").n *(f.((d+c).(2*n)) - f.(c.(2*n))) by A2,A3,FUNCT_2:108
      .= (h1").n *(f.(d.(2*n)+c.(2*n)) - f.(c.(2*n))) by SEQ_1:7
      .= (h1").n *(f.(h1.n+c.(2*n)) - f.(c.(2*n))) by A8
      .= (h1").n *(f.(h1.n+c.(2*n)) - f.(c.n)) by VALUED_0:23
      .= (h1").n *(f.(h1.n+c.n) - f.(c.n)) by VALUED_0:23
      .= (h1").n *(f.((h1+c).n) - f.(c.n)) by SEQ_1:7
      .= (h1").n *((f/*(h1+c)).n - f.(c.n)) by A4,FUNCT_2:108
      .= (h1").n *((f/*(h1+c)).n - (f/*c).n) by A2,A3,FUNCT_2:108
      .= (h1").n *(f/*(h1+c) - f/*c).n by RFUNCT_2:1
      .= (h1"(#)(f/*(h1+c) - f/*c)).n by SEQ_1:8;
  end;
  then h1"(#)(f/*(h1+c) - f/*c) is subsequence of d"(#)(f/*(d+c) - f/*c) by
FUNCT_2:63;
  then lim (h1"(#)(f/*(h1+c) - f/*c)) = lim (d"(#) (f/*(d+c) -f/*c)) by A19,
SEQ_4:17;
  hence thesis by A19,A17,SEQ_4:17;
end;
