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

theorem Th33:
  for L be lower-bounded modular LATTICE for S being ExtensionSeq2
  of the carrier of L, BasicDF(L) for FS being non empty set for FD being
distance_function of FS,L for x,y being Element of FS for a,b being Element of
L st FS = union the set of all  (S.i)`1 where i is Element of NAT & FD =
  union the set of all  (S.i)`2 where i is Element of NAT & FD.(x,y) <= a
  "\/"b ex z1,z2 being Element of FS st FD.(x,z1) = a & FD.(z1,z2) = (FD.(x,y)
  "\/"a)"/\"b & FD.(z2,y) = a
proof
  let L be lower-bounded modular LATTICE;
  let S be ExtensionSeq2 of the carrier of L, BasicDF(L);
  let FS be non empty set, FD be distance_function of FS,L;
  let x,y be Element of FS;
  let a,b be Element of L;
  assume that
A1: FS = union the set of all  (S.i)`1 where i is Element of NAT and
A2: FD = union the set of all  (S.i)`2 where i is Element of NAT and
A3: FD.(x,y) <= a"\/"b;
  (S.0)`1 in the set of all  (S.i)`1 where i is Element of NAT;
  then reconsider
  X = the set of all  (S.i)`1 where i is Element of NAT as
  non empty set;
  consider x1 being set such that
A4: x in x1 and
A5: x1 in X by A1,TARSKI:def 4;
  consider k being Element of NAT such that
A6: x1 = (S.k)`1 by A5;
  consider A1 being non empty set, d1 being distance_function of A1,L, Aq1
  being non empty set, dq1 being distance_function of Aq1,L such that
A7: Aq1, dq1 is_extension2_of A1,d1 and
A8: S.k = [A1,d1] and
A9: S.(k+1) = [Aq1,dq1] by Def11;
A10: [A1,d1]`1 = A1;
A11: x in A1 by A4,A6,A8;
  [A1,d1]`2 = d1;
  then d1 in the set of all  (S.i)`2 where i is Element of NAT by A8;
  then
A12: d1 c= FD by A2,ZFMISC_1:74;
A13: [Aq1,dq1]`1 = Aq1;
  then Aq1 in the set of all  (S.i)`1 where i is Element of NAT by A9;
  then
A14: Aq1 c= FS by A1,ZFMISC_1:74;
  [Aq1,dq1]`2 = dq1;
  then dq1 in the set of all  (S.i)`2 where i is Element of NAT by A9;
  then
A15: dq1 c= FD by A2,ZFMISC_1:74;
  consider y1 being set such that
A16: y in y1 and
A17: y1 in X by A1,TARSKI:def 4;
  consider l being Element of NAT such that
A18: y1 = (S.l)`1 by A17;
  consider A2 being non empty set, d2 being distance_function of A2,L, Aq2
  being non empty set, dq2 being distance_function of Aq2,L such that
A19: Aq2, dq2 is_extension2_of A2,d2 and
A20: S.l = [A2,d2] and
A21: S.(l+1) = [Aq2,dq2] by Def11;
A22: [A2,d2]`1 = A2;
A23: y in A2 by A16,A18,A20;
  [A2,d2]`2 = d2;
  then d2 in the set of all  (S.i)`2 where i is Element of NAT by A20;
  then
A24: d2 c= FD by A2,ZFMISC_1:74;
A25: [Aq2,dq2]`1 = Aq2;
  then Aq2 in the set of all  (S.i)`1 where i is Element of NAT by A21;
  then
A26: Aq2 c= FS by A1,ZFMISC_1:74;
  [Aq2,dq2]`2 = dq2;
  then dq2 in the set of all  (S.i)`2 where i is Element of NAT by A21;
  then
A27: dq2 c= FD by A2,ZFMISC_1:74;
  per cases;
  suppose
    k <= l;
    then A1 c= A2 by A10,A22,Th30,A8,A20;
    then reconsider x9=x,y9=y as Element of A2
      by A11,A16,A18,A20;
    A2 c= Aq2 by A22,A25,Th30,A20,A21,NAT_1:11;
    then reconsider x99 = x9,y99 = y9 as Element of Aq2;
    dom d2 = [:A2,A2:] by FUNCT_2:def 1;
    then
A28: FD.(x,y) = d2.[x9,y9] by A24,GRFUNC_1:2
      .= d2.(x9,y9);
    then consider z1,z2 being Element of Aq2 such that
A29: dq2.(x,z1) = a and
A30: dq2.(z1,z2) = (d2.(x9,y9)"\/"a)"/\"b and
A31: dq2.(z2,y) = a by A3,A19,Th29;
    reconsider z19 = z1, z29 = z2 as Element of FS by A26;
    take z19,z29;
A32: dom dq2 = [:Aq2,Aq2:] by FUNCT_2:def 1;
    hence FD.(x,z19) = dq2.[x99,z1] by A27,GRFUNC_1:2
      .= a by A29;
    thus FD.(z19,z29) = dq2.[z1,z2] by A27,A32,GRFUNC_1:2
      .= (FD.(x,y)"\/"a)"/\"b by A28,A30;
    thus FD.(z29,y) = dq2.[z2,y99] by A27,A32,GRFUNC_1:2
      .= a by A31;
  end;
  suppose
    l <= k;
    then A2 c= A1 by A10,A22,Th30,A8,A20;
    then reconsider x9=x,y9=y as Element of A1 by A4,A6,A8,A23;
    A1 c= Aq1 by A10,A13,Th30,A8,A9,NAT_1:11;
    then reconsider x99 = x9,y99 = y9 as Element of Aq1;
    dom d1 = [:A1,A1:] by FUNCT_2:def 1;
    then
A33: FD.(x,y) = d1.[x9,y9] by A12,GRFUNC_1:2
      .= d1.(x9,y9);
    then consider z1,z2 being Element of Aq1 such that
A34: dq1.(x,z1) = a and
A35: dq1.(z1,z2) = (d1.(x9,y9)"\/"a)"/\"b and
A36: dq1.(z2,y) = a by A3,A7,Th29;
    reconsider z19 = z1, z29 = z2 as Element of FS by A14;
    take z19,z29;
A37: dom dq1 = [:Aq1,Aq1:] by FUNCT_2:def 1;
    hence FD.(x,z19) = dq1.[x99,z1] by A15,GRFUNC_1:2
      .= a by A34;
    thus FD.(z19,z29) = dq1.[z1,z2] by A15,A37,GRFUNC_1:2
      .= (FD.(x,y)"\/"a)"/\"b by A33,A35;
    thus FD.(z29,y) = dq1.[z2,y99] by A15,A37,GRFUNC_1:2
      .= a by A36;
  end;
end;
