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 Th15:
  dom(F1.0) = dom(F2.0) & F1 is with_the_same_dom & (for n be Nat
  holds F1.n = - F2.n) implies lim_inf F1 = -lim_sup F2 & lim_sup F1 = -lim_inf
  F2
proof
  assume that
A1: dom(F1.0) = dom(F2.0) and
A2: F1 is with_the_same_dom and
A3: for n be Nat holds F1.n = - F2.n;
A4: dom lim_inf F1 = dom(F1.0) by MESFUNC8:def 7;
A5: dom lim_sup F2 = dom(F2.0) by MESFUNC8:def 8;
A6: now
    let x be Element of X;
    assume
A7: x in dom(F1.0);
    let n be Nat;
    dom(F1.n) = dom(F1.0) by A2,MESFUNC8:def 2;
    then
A8: x in dom(-F2.n) by A3,A7;
    (F1.n).x = (-F2.n).x by A3;
    then (F1.n).x = -(F2.n).x by A8,MESFUNC1:def 7;
    then (F1#x).n = -(F2.n).x by MESFUNC5:def 13;
    hence (F2#x).n = -(F1#x).n by MESFUNC5:def 13;
  end;
A9: now
    let x be Element of X;
    assume
A10: x in dom lim_inf F1;
    then
A11: (lim_inf F1).x = lim_inf(F1#x) by MESFUNC8:def 7;
    x in dom(-lim_sup F2) by A1,A4,A5,A10,MESFUNC1:def 7;
    then
A12: (-lim_sup F2).x = -(lim_sup F2).x by MESFUNC1:def 7;
A13: for n be Nat holds (F2#x).n = -(F1#x).n by A4,A6,A10;
    (lim_sup F2).x = lim_sup(F2#x) by A1,A4,A5,A10,MESFUNC8:def 8;
    hence (lim_inf F1).x = (-lim_sup F2).x by A13,A11,A12,Th14;
  end;
  dom(-lim_sup F2) = dom lim_sup F2 by MESFUNC1:def 7;
  hence lim_inf F1 = -lim_sup F2 by A1,A4,A5,A9,PARTFUN1:5;
A14: dom lim_sup F1 = dom(F1.0) by MESFUNC8:def 8;
A15: dom lim_inf F2 = dom(F2.0) by MESFUNC8:def 7;
A16: for x be Element of X st x in dom lim_sup F1 holds (lim_sup F1).x = (-
  lim_inf F2).x
  proof
    let x be Element of X;
    assume
A17: x in dom lim_sup F1;
    then
A18: (lim_sup F1).x = lim_sup(F1#x) by MESFUNC8:def 8;
    x in dom(-lim_inf F2) by A1,A14,A15,A17,MESFUNC1:def 7;
    then
A19: (-lim_inf F2).x = -(lim_inf F2).x by MESFUNC1:def 7;
A20: for n be Nat holds (F2#x).n = -(F1#x).n by A14,A6,A17;
    (lim_inf F2).x = lim_inf(F2#x) by A1,A14,A15,A17,MESFUNC8:def 7;
    hence thesis by A20,A18,A19,Th14;
  end;
  dom(-lim_inf F2) = dom lim_inf F2 by MESFUNC1:def 7;
  hence thesis by A1,A14,A15,A16,PARTFUN1:5;
end;
