reserve x for set,
  C for Ordinal,
  L0 for Sequence;
reserve O1,O2 for Ordinal;

theorem Th25:
  for A being non empty set for L be lower-bounded LATTICE for d
  be BiFunction of A,L st d is zeroed for q being QuadrSeq of d for O being
  Ordinal holds ConsecutiveDelta2(q,O) is zeroed
proof
  let A be non empty set;
  let L be lower-bounded LATTICE;
  let d be BiFunction of A,L;
  assume
A1: d is zeroed;
  let q be QuadrSeq of d;
  let O be Ordinal;
  defpred X[Ordinal] means ConsecutiveDelta2(q,$1) is zeroed;
A2: for O1 being Ordinal st X[O1] holds X[succ O1]
  proof
    let O1 be Ordinal;
    assume ConsecutiveDelta2(q,O1) is zeroed;
    then
A3: new_bi_fun2(ConsecutiveDelta2(q,O1),Quadr2(q,O1)) is zeroed by Th10;
    let z be Element of ConsecutiveSet2(A,succ O1);
    reconsider z9 = z as Element of new_set2 ConsecutiveSet2(A,O1) by Th15;
    ConsecutiveDelta2(q,succ O1) = new_bi_fun2(BiFun(ConsecutiveDelta2(q,
    O1 ) , ConsecutiveSet2(A,O1),L),Quadr2(q,O1)) by Th19
      .= new_bi_fun2(ConsecutiveDelta2(q,O1),Quadr2(q,O1)) by LATTICE5:def 15;
    hence
    ConsecutiveDelta2(q,succ O1).(z,z) = new_bi_fun2(ConsecutiveDelta2(q,
    O1),Quadr2(q,O1)).(z9,z9)
      .= Bottom L by A3;
  end;
A4: for O1 st O1 <> 0 & O1 is limit_ordinal & for O2 st O2 in O1 holds X[O2
  ] holds X[O1]
  proof
    deffunc U(Ordinal) = ConsecutiveDelta2(q,$1);
    let O2 be Ordinal;
    assume that
A5: O2 <> 0 & O2 is limit_ordinal and
A6: for O1 be Ordinal st O1 in O2 holds ConsecutiveDelta2(q,O1) is zeroed;
    set CS = ConsecutiveSet2(A,O2);
    consider Ls being Sequence such that
A7: dom Ls = O2 & for O1 being Ordinal st O1 in O2 holds Ls.O1 = U(O1
    ) from ORDINAL2:sch 2;
    ConsecutiveDelta2(q,O2) = union rng Ls by A5,A7,Th20;
    then reconsider f = union rng Ls as BiFunction of CS,L;
    deffunc U(Ordinal) = ConsecutiveSet2(A,$1);
    consider Ts being Sequence such that
A8: dom Ts = O2 & for O1 being Ordinal st O1 in O2 holds Ts.O1 = U(O1
    ) from ORDINAL2:sch 2;
A9: ConsecutiveSet2(A,O2) = union rng Ts by A5,A8,Th16;
    f is zeroed
    proof
      let x be Element of CS;
      consider y being set such that
A10:  x in y and
A11:  y in rng Ts by A9,TARSKI:def 4;
      consider o being object such that
A12:  o in dom Ts and
A13:  y = Ts.o by A11,FUNCT_1:def 3;
      reconsider o as Ordinal by A12;
A14:  Ls.o = ConsecutiveDelta2(q,o) by A7,A8,A12;
      then reconsider h = Ls.o as BiFunction of ConsecutiveSet2(A,o),L;
      reconsider x9 = x as Element of ConsecutiveSet2(A,o) by A8,A10,A12,A13;
A15:  dom h = [:ConsecutiveSet2(A,o),ConsecutiveSet2(A,o):] by FUNCT_2:def 1;
A16:  h is zeroed
      proof
        let z be Element of ConsecutiveSet2(A,o);
A17:    ConsecutiveDelta2(q,o) is zeroed by A6,A8,A12;
        thus h.(z,z) = ConsecutiveDelta2(q,o).(z,z) by A7,A8,A12
          .= Bottom L by A17;
      end;
      ConsecutiveDelta2(q,o) in rng Ls by A7,A8,A12,A14,FUNCT_1:def 3;
      then
A18:  h c= f by A14,ZFMISC_1:74;
      x in ConsecutiveSet2(A,o) by A8,A10,A12,A13;
      then [x,x] in dom h by A15,ZFMISC_1:87;
      hence f.(x,x) = h.(x9,x9) by A18,GRFUNC_1:2
        .= Bottom L by A16;
    end;
    hence thesis by A5,A7,Th20;
  end;
A19: X[0]
  proof
    let z be Element of ConsecutiveSet2(A,0);
    reconsider z9 = z as Element of A by Th14;
    thus ConsecutiveDelta2(q,0).(z,z) = d.(z9,z9) by Th18
      .= Bottom L by A1;
  end;
  for O being Ordinal holds X[O] from ORDINAL2:sch 1(A19,A2,A4);
  hence thesis;
end;
