reserve X for non empty set;
reserve e,e1,e2,e19,e29 for Equivalence_Relation of X,
  x,y,x9,y9 for set;
reserve A for non empty set,
  L for lower-bounded LATTICE;
reserve T,L1 for Sequence,
  O,O1,O2,O3,C for Ordinal;

theorem Th30:
  for d be BiFunction of A,L for q being QuadrSeq of d holds
  ConsecutiveDelta(q,O) is BiFunction of ConsecutiveSet(A,O),L
proof
  let d be BiFunction of A,L;
  let q be QuadrSeq of d;
  defpred Y[Ordinal] means ConsecutiveDelta(q,$1) is BiFunction of
  ConsecutiveSet(A,$1),L;
A1: for O1 st Y[O1] holds Y[succ O1]
  proof
    let O1;
    assume ConsecutiveDelta(q,O1) is BiFunction of ConsecutiveSet(A,O1),L;
    then reconsider
    CD = ConsecutiveDelta(q,O1) as BiFunction of ConsecutiveSet(A,
    O1),L;
A2: ConsecutiveSet(A,succ O1) = new_set ConsecutiveSet(A,O1) by Th22;
    ConsecutiveDelta(q,succ O1) = new_bi_fun(BiFun(ConsecutiveDelta(q,O1),
    ConsecutiveSet(A,O1),L),Quadr(q,O1)) by Th27
      .= new_bi_fun(CD,Quadr(q,O1)) by Def15;
    hence thesis by A2;
  end;
A3: for O1 st O1 <> 0 & O1 is limit_ordinal & for O2 st O2 in O1 holds Y[O2]
    holds Y[O1]
  proof
    deffunc U(Ordinal) = ConsecutiveDelta(q,$1);
    let O1;
    assume that
A4: O1 <> 0 and
A5: O1 is limit_ordinal and
A6: for O2 st O2 in O1 holds ConsecutiveDelta(q,O2) is BiFunction of
    ConsecutiveSet(A,O2),L;
    consider Ls being Sequence such that
A7: dom Ls = O1 & for O2 being Ordinal st O2 in O1 holds Ls.O2 = U(O2
    ) from ORDINAL2:sch 2;
A8: for O,O2 st O c= O2 & O2 in dom Ls holds Ls.O c= Ls.O2
    proof
      let O;
      defpred X[Ordinal] means O c= $1 & $1 in dom Ls implies Ls.O c= Ls.$1;
A9:   for O2 st O2 <> 0 & O2 is limit_ordinal & for O3 st O3 in O2 holds
      X[O3] holds X[O2]
      proof
        deffunc U(Ordinal) = ConsecutiveDelta(q,$1);
        let O2;
        assume that
A10:    O2 <> 0 & O2 is limit_ordinal and
        for O3 st O3 in O2 holds O c= O3 & O3 in dom Ls implies Ls.O c= Ls
        . O3;
        assume that
A11:    O c= O2 and
A12:    O2 in dom Ls;
        consider Lt being Sequence such that
A13:    dom Lt = O2 & for O3 being Ordinal st O3 in O2 holds Lt.O3 =
        U(O3) from ORDINAL2:sch 2;
A14:    Ls.O2 = ConsecutiveDelta(q,O2) by A7,A12
          .= union rng Lt by A10,A13,Th28;
        per cases;
        suppose
          O = O2;
          hence thesis;
        end;
        suppose
          O <> O2;
          then
A15:      O c< O2 by A11;
          then
A16:      O in O2 by ORDINAL1:11;
          then Ls.O = ConsecutiveDelta(q,O) by A7,A12,ORDINAL1:10
            .= Lt.O by A13,A15,ORDINAL1:11;
          then Ls.O in rng Lt by A13,A16,FUNCT_1:def 3;
          hence thesis by A14,ZFMISC_1:74;
        end;
      end;
A17:  for O2 st X[O2] holds X[succ O2]
      proof
        let O2;
        assume
A18:    O c= O2 & O2 in dom Ls implies Ls.O c= Ls.O2;
        assume that
A19:    O c= succ O2 and
A20:    succ O2 in dom Ls;
        per cases;
        suppose
          O = succ O2;
          hence thesis;
        end;
        suppose
          O <> succ O2;
          then O c< succ O2 by A19;
          then
A21:      O in succ O2 by ORDINAL1:11;
A22:      O2 in succ O2 by ORDINAL1:6;
          then O2 in dom Ls by A20,ORDINAL1:10;
          then reconsider cd2 = ConsecutiveDelta(q,O2) as BiFunction of
          ConsecutiveSet(A,O2),L by A6,A7;
          Ls.succ O2 = ConsecutiveDelta(q,succ O2) by A7,A20
            .= new_bi_fun(BiFun(ConsecutiveDelta(q,O2), ConsecutiveSet(A,O2)
          ,L),Quadr(q,O2)) by Th27
            .= new_bi_fun(cd2,Quadr(q,O2)) by Def15;
          then ConsecutiveDelta(q,O2) c= Ls.succ O2 by Th19;
          then Ls.O2 c= Ls.succ O2 by A7,A20,A22,ORDINAL1:10;
          hence thesis by A18,A20,A21,A22,ORDINAL1:10,22;
        end;
      end;
A23:  X[0];
      thus for O2 holds X[O2] from ORDINAL2:sch 1(A23,A17,A9);
    end;
    for x,y being set st x in rng Ls & y in rng Ls holds x,y are_c=-comparable
    proof
      let x,y be set;
      assume that
A24:  x in rng Ls and
A25:  y in rng Ls;
      consider o1 being object such that
A26:  o1 in dom Ls and
A27:  Ls.o1 = x by A24,FUNCT_1:def 3;
      consider o2 being object such that
A28:  o2 in dom Ls and
A29:  Ls.o2 = y by A25,FUNCT_1:def 3;
      reconsider o19 = o1, o29 = o2 as Ordinal by A26,A28;
      o19 c= o29 or o29 c= o19;
      then x c= y or y c= x by A8,A26,A27,A28,A29;
      hence thesis;
    end;
    then
A30: rng Ls is c=-linear;
    set Y = the carrier of L, X = [:ConsecutiveSet(
    A,O1),ConsecutiveSet(A,O1):], f = union rng Ls;
    rng Ls c= PFuncs(X,Y)
    proof
      let z be object;
      assume z in rng Ls;
      then consider o being object such that
A31:  o in dom Ls and
A32:  z = Ls.o by FUNCT_1:def 3;
      reconsider o as Ordinal by A31;
      Ls.o = ConsecutiveDelta(q,o) by A7,A31;
      then reconsider
      h = Ls.o as BiFunction of ConsecutiveSet(A,o),L by A6,A7,A31;
      o c= O1 by A7,A31,ORDINAL1:def 2;
      then dom h = [:ConsecutiveSet(A,o),ConsecutiveSet(A,o):] &
      ConsecutiveSet(A,o) c= ConsecutiveSet(A,O1) by Th29,FUNCT_2:def 1;
      then rng h c= Y & dom h c= X by ZFMISC_1:96;
      hence thesis by A32,PARTFUN1:def 3;
    end;
    then f in PFuncs(X,Y) by A30,TREES_2:40;
    then
A33: ex g being Function st f = g & dom g c= X & rng g c= Y by PARTFUN1:def 3;
    reconsider o1 = O1 as non empty Ordinal by A4;
    set YY = the set of all
 [:ConsecutiveSet(A,O2),ConsecutiveSet(A,O2):] where O2 is
    Element of o1 ;
    deffunc U(Ordinal) = ConsecutiveSet(A,$1);
    consider Ts being Sequence such that
A34: dom Ts = O1 & for O2 being Ordinal st O2 in O1 holds Ts.O2 = U(O2
    ) from ORDINAL2:sch 2;
    Ls is Function-yielding
    proof
      let x be object;
      assume
A35:  x in dom Ls;
      then reconsider o = x as Ordinal;
      Ls.o = ConsecutiveDelta(q,o) by A7,A35;
      hence thesis by A6,A7,A35;
    end;
    then reconsider LsF = Ls as Function-yielding Function;
A36: rng doms LsF = YY
    proof
      thus rng doms LsF c= YY
      proof
        let Z be object;
        assume Z in rng doms LsF;
        then consider o being object such that
A37:    o in dom doms LsF and
A38:    Z = (doms LsF).o by FUNCT_1:def 3;
A39:    o in dom LsF by A37,FUNCT_6:59;
        then reconsider o9 = o as Element of o1 by A7;
        Ls.o9 = ConsecutiveDelta(q,o9) by A7;
        then reconsider
        ls = Ls.o9 as BiFunction of ConsecutiveSet(A,o9),L by A6;
        Z = dom ls by A38,A39,FUNCT_6:22
          .= [:ConsecutiveSet(A,o9),ConsecutiveSet(A,o9):] by FUNCT_2:def 1;
        hence thesis;
      end;
      let Z be object;
      assume Z in YY;
      then consider o being Element of o1 such that
A40:  Z = [:ConsecutiveSet(A,o),ConsecutiveSet(A,o):];
      Ls.o = ConsecutiveDelta(q,o) by A7;
      then reconsider ls = Ls.o as BiFunction of ConsecutiveSet(A,o),L by A6;
      o in dom LsF by A7;
      then
A41:  o in dom doms LsF by FUNCT_6:59;
      Z = dom ls by A40,FUNCT_2:def 1
        .= (doms LsF).o by A7,FUNCT_6:22;
      hence thesis by A41,FUNCT_1:def 3;
    end;
    {} in O1 by A4,ORDINAL3:8;
    then reconsider RTs = rng Ts as non empty set by A34,FUNCT_1:3;
    reconsider f as Function by A33;
A42: dom f = union rng doms LsF by Th1;
A43: YY = { [:a,a:] where a is Element of RTs : a in RTs }
    proof
      set XX = { [:a,a:] where a is Element of RTs : a in RTs };
      thus YY c= XX
      proof
        let Z be object;
        assume Z in YY;
        then consider o being Element of o1 such that
A44:    Z = [:ConsecutiveSet(A,o),ConsecutiveSet(A,o):];
        Ts.o = ConsecutiveSet(A,o) by A34;
        then reconsider CoS = ConsecutiveSet(A,o) as Element of RTs by A34,
FUNCT_1:def 3;
        Z = [:CoS,CoS:] by A44;
        hence thesis;
      end;
      let Z be object;
      assume Z in XX;
      then consider a being Element of RTs such that
A45:  Z = [:a,a:] and
      a in RTs;
      consider o being object such that
A46:  o in dom Ts and
A47:  a = Ts.o by FUNCT_1:def 3;
      reconsider o9 = o as Ordinal by A46;
      a = ConsecutiveSet(A,o9) by A34,A46,A47;
      hence thesis by A34,A45,A46;
    end;
    for x,y being set st x in RTs & y in RTs holds x,y are_c=-comparable
    proof
      let x,y be set;
      assume that
A48:  x in RTs and
A49:  y in RTs;
      consider o1 being object such that
A50:  o1 in dom Ts and
A51:  Ts.o1 = x by A48,FUNCT_1:def 3;
      consider o2 being object such that
A52:  o2 in dom Ts and
A53:  Ts.o2 = y by A49,FUNCT_1:def 3;
      reconsider o19 = o1, o29 = o2 as Ordinal by A50,A52;
A54:  Ts.o29 = ConsecutiveSet(A,o29) by A34,A52;
A55:  o19 c= o29 or o29 c= o19;
      Ts.o19 = ConsecutiveSet(A,o19) by A34,A50;
      then x c= y or y c= x by A51,A53,A54,A55,Th29;
      hence thesis;
    end;
    then
A56: RTs is c=-linear;
A57: ConsecutiveDelta(q,O1) = union rng Ls by A4,A5,A7,Th28;
    X = [:union rng Ts, ConsecutiveSet(A,O1):] by A4,A5,A34,Th23
      .= [:union RTs, union RTs :] by A4,A5,A34,Th23
      .= dom f by A42,A36,A56,A43,Th3;
    hence thesis by A57,A33,FUNCT_2:def 1,RELSET_1:4;
  end;
  ConsecutiveSet(A,{}) = A by Th21;
  then
A58: Y[0] by Th26;
  for O holds Y[O] from ORDINAL2:sch 1(A58,A1,A3);
  hence thesis;
end;
