reserve X for non empty set,
  F for with_the_same_dom Functional_Sequence of X, ExtREAL,
  seq,seq1,seq2 for ExtREAL_sequence,
  x for Element of X,
  a,r for R_eal,
  n,m,k for Nat;
reserve S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S;
reserve F1,F2 for Functional_Sequence of X,ExtREAL,
  f,g,P for PartFunc of X, ExtREAL;

theorem Th14:
  (for n be Nat holds seq1.n = -seq2.n) implies lim_inf seq2 = -
  lim_sup seq1 & lim_sup seq2 = -lim_inf seq1
proof
  assume
A1: for n be Nat holds seq1.n = -seq2.n;
  now
    let z be object;
    assume z in rng (inferior_realsequence seq2);
    then consider n be object such that
A2: n in dom (inferior_realsequence seq2) and
A3: z = (inferior_realsequence seq2).n by FUNCT_1:def 3;
    reconsider n as Element of NAT by A2;
    consider R2 be non empty Subset of ExtREAL such that
A4: R2 = {seq2.k where k is Nat : n <= k} and
A5: z = inf R2 by A3,RINFSUP2:def 6;
    reconsider z2 = z as Element of ExtREAL by A3, XXREAL_0:def 1;
    set R1 = {seq1.k where k is Nat : n <= k};
    reconsider R1 as non empty Subset of ExtREAL by RINFSUP2:5;
    set z1 = -z2;
A6: ex K1 be non empty Subset of ExtREAL st K1 = {seq1.k where k is
Nat: n <= k} & (superior_realsequence seq1).n = sup K1 by
RINFSUP2:def 7;
    now
      let x be object;
      assume x in R1;
      then consider k be Nat such that
A7:   x = seq1.k and
A8:   n <= k;
      reconsider x1=x as Element of ExtREAL by A7;
      -x1 = -(-seq2.k) by A1,A7;
      then -x1 in {seq2.k2 where k2 is Nat : n <= k2} by A8;
      then -(-x1) in -R2 by A4;
      hence x in -R2;
    end;
    then
A9: R1 c= -R2 by TARSKI:def 3;
    now
      let x be object;
      assume x in -R2;
      then consider y be R_eal such that
A10:  x = -y and
A11:  y in R2;
      consider k be Nat such that
A12:  y = seq2.k and
A13:  n <= k by A4,A11;
      seq1.k = -seq2.k by A1;
      hence x in R1 by A10,A12,A13;
    end;
    then -R2 c= R1 by TARSKI:def 3;
    then -R2 = R1 by A9,XBOOLE_0:def 10;
    then (superior_realsequence seq1).n = z1 by A5,A6,SUPINF_2:15;
    then
A14: z1 in rng(superior_realsequence seq1) by FUNCT_2:4;
    z2 = -z1;
    hence z in -rng(superior_realsequence seq1) by A14;
  end;
  then
A15: rng (inferior_realsequence seq2) c= -(rng (superior_realsequence seq1))
  by TARSKI:def 3;
  now
    let z be object;
    assume z in rng (superior_realsequence seq2);
    then consider n be object such that
A16: n in dom (superior_realsequence seq2) and
A17: z = (superior_realsequence seq2).n by FUNCT_1:def 3;
    reconsider n as Element of NAT by A16;
    consider R2 be non empty Subset of ExtREAL such that
A18: R2 = {seq2.k where k is Nat : n <= k} and
A19: z = sup R2 by A17,RINFSUP2:def 7;
    reconsider z2 = z as Element of ExtREAL by A17,XXREAL_0:def 1;
    set R1 = {seq1.k where k is Nat : n <= k};
    reconsider R1 as non empty Subset of ExtREAL by RINFSUP2:5;
    set z1 = -z2;
A20: ex K1 be non empty Subset of ExtREAL st K1 = {seq1.k where k is
Nat: n <= k} & (inferior_realsequence seq1).n = inf K1 by
RINFSUP2:def 6;
    now
      let x be object;
      assume x in R1;
      then consider k be Nat such that
A21:  x = seq1.k and
A22:  n <= k;
      reconsider x1=x as Element of ExtREAL by A21;
      -x1 = -(-seq2.k) by A1,A21;
      then -x1 in {seq2.k2 where k2 is Nat : n <= k2} by A22;
      then -(-x1) in -R2 by A18;
      hence x in -R2;
    end;
    then
A23: R1 c= -R2 by TARSKI:def 3;
    now
      let x be object;
      assume x in -R2;
      then consider y be R_eal such that
A24:  x = -y and
A25:  y in R2;
      consider k be Nat such that
A26:  y = seq2.k and
A27:  n <= k by A18,A25;
      seq1.k = -seq2.k by A1;
      hence x in R1 by A24,A26,A27;
    end;
    then -R2 c= R1 by TARSKI:def 3;
    then -R2 = R1 by A23,XBOOLE_0:def 10;
    then (inferior_realsequence seq1).n = z1 by A19,A20,SUPINF_2:14;
    then
A28: z1 in rng(inferior_realsequence seq1) by FUNCT_2:4;
    z2 = -z1;
    hence z in -rng(inferior_realsequence seq1) by A28;
  end;
  then
A29: rng (superior_realsequence seq2) c= -(rng (inferior_realsequence seq1))
  by TARSKI:def 3;
  now
    let z be object;
    assume z in -(rng (superior_realsequence seq1));
    then consider t be R_eal such that
A30: z = -t and
A31: t in rng (superior_realsequence seq1);
    consider n be object such that
A32: n in dom (superior_realsequence seq1) and
A33: t = (superior_realsequence seq1).n by A31,FUNCT_1:def 3;
    reconsider n as Element of NAT by A32;
    consider R1 be non empty Subset of ExtREAL such that
A34: R1 = {seq1.k where k is Nat : n <= k} and
A35: t = sup R1 by A33,RINFSUP2:def 7;
    reconsider z1 = z as Element of ExtREAL by A30;
    set R2 = {seq2.k where k is Nat : n <= k};
    reconsider R2 as non empty Subset of ExtREAL by RINFSUP2:5;
A36: ex K2 be non empty Subset of ExtREAL st K2 = {seq2.k where k is
Nat: n <= k} & (inferior_realsequence seq2).n = inf K2 by
RINFSUP2:def 6;
    now
      let x be object;
      assume x in R2;
      then consider k be Nat such that
A37:  x = seq2.k and
A38:  n <= k;
      reconsider x1=x as Element of ExtREAL by A37;
      -x1 = -(-seq1.k) by A1,A37;
      then -x1 in {seq1.k2 where k2 is Nat : n <= k2} by A38;
      then -(-x1) in -R1 by A34;
      hence x in -R1;
    end;
    then
A39: R2 c= -R1 by TARSKI:def 3;
    now
      let x be object;
      assume x in -R1;
      then consider y be R_eal such that
A40:  x = -y and
A41:  y in R1;
      consider k be Nat such that
A42:  y = seq1.k and
A43:  n <= k by A34,A41;
      seq1.k = -seq2.k by A1;
      hence x in R2 by A40,A42,A43;
    end;
    then -R1 c= R2 by TARSKI:def 3;
    then -R1 = R2 by A39,XBOOLE_0:def 10;
    then (inferior_realsequence seq2).n = z1 by A30,A35,A36,SUPINF_2:14;
    hence z in rng (inferior_realsequence seq2) by FUNCT_2:4;
  end;
  then
  -(rng (superior_realsequence seq1)) c= rng (inferior_realsequence seq2)
  by TARSKI:def 3;
  then rng (inferior_realsequence seq2) = -(rng (superior_realsequence seq1))
  by A15,XBOOLE_0:def 10;
  hence lim_inf seq2 = - lim_sup seq1 by SUPINF_2:15;
  now
    let z be object;
    assume z in -(rng (inferior_realsequence seq1));
    then consider t be R_eal such that
A44: z = -t and
A45: t in rng (inferior_realsequence seq1);
    consider n be object such that
A46: n in dom (inferior_realsequence seq1) and
A47: t = (inferior_realsequence seq1).n by A45,FUNCT_1:def 3;
    reconsider n as Element of NAT by A46;
    consider R1 be non empty Subset of ExtREAL such that
A48: R1 = {seq1.k where k is Nat : n <= k} and
A49: t = inf R1 by A47,RINFSUP2:def 6;
    reconsider z1 = z as Element of ExtREAL by A44;
    set R2 = {seq2.k where k is Nat : n <= k};
    reconsider R2 as non empty Subset of ExtREAL by RINFSUP2:5;
A50: ex K2 be non empty Subset of ExtREAL st K2 = {seq2.k where k is
Nat: n <= k} & (superior_realsequence seq2).n = sup K2 by
RINFSUP2:def 7;
    now
      let x be object;
      assume x in R2;
      then consider k be Nat such that
A51:  x = seq2.k and
A52:  n <= k;
      reconsider x1=x as Element of ExtREAL by A51;
      seq1.k = -seq2.k by A1;
      then -x1 in R1 by A48,A51,A52;
      then -(-x1) in -R1;
      hence x in -R1;
    end;
    then
A53: R2 c= -R1 by TARSKI:def 3;
    now
      let x be object;
      assume x in -R1;
      then consider y be R_eal such that
A54:  x = -y and
A55:  y in R1;
      consider k be Nat such that
A56:  y = seq1.k and
A57:  n <= k by A48,A55;
      seq1.k = -seq2.k by A1;
      hence x in R2 by A54,A56,A57;
    end;
    then -R1 c= R2 by TARSKI:def 3;
    then -R1 = R2 by A53,XBOOLE_0:def 10;
    then (superior_realsequence seq2).n = z1 by A44,A49,A50,SUPINF_2:15;
    hence z in rng (superior_realsequence seq2) by FUNCT_2:4;
  end;
  then -(rng (inferior_realsequence seq1)) c= rng (superior_realsequence seq2
  ) by TARSKI:def 3;
  then rng (superior_realsequence seq2) = -(rng (inferior_realsequence seq1))
  by A29,XBOOLE_0:def 10;
  hence thesis by SUPINF_2:14;
end;
