reserve X for non empty set;
reserve e,e1,e2,e19,e29 for Equivalence_Relation of X,
  x,y,x9,y9 for set;

theorem Th13:
  for Y being Sublattice of EqRelLATT X, n being Element of NAT st
  (ex e st e in the carrier of Y & e <> id X) &
  (for e1,e2 for x,y being object st e1 in the
  carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 holds (ex F being
  non empty FinSequence of X st len F = n+2 & x,y are_joint_by F,e1,e2)) holds
  type_of Y <= n
proof
  let Y be Sublattice of EqRelLATT X, n be Element of NAT;
  assume that
A1: ex e st e in the carrier of Y & e <> id X and
A2: for e1,e2 for x,y being object
   st e1 in the carrier of Y & e2 in the carrier of Y & [
x,y] in e1 "\/" e2 holds ex F being non empty FinSequence of X st len F = n+2 &
  x,y are_joint_by F,e1,e2 and
A3: n < type_of Y;
  n+1 <= type_of Y by A3,NAT_1:13;
  then consider m being Nat such that
A4: type_of Y = (n+1)+m by NAT_1:10;
  reconsider m as Element of NAT by ORDINAL1:def 12;
  n+1+m+1 = n+m+2;
  then consider e1,e2 being Equivalence_Relation of X,x,y being object
   such that
A5: e1 in the carrier of Y & e2 in the carrier of Y & [x,y] in e1 "\/" e2 and
A6: not (ex F being non empty FinSequence of X st len F = (n+m)+2 & x,y
  are_joint_by F,e1,e2) by A1,A4,Def4;
A7: n+2+m = (n+m)+2;
  field e2 = X by EQREL_1:9;
  then
A8: e2 is_reflexive_in X by RELAT_2:def 9;
  field e1 = X by EQREL_1:9;
  then
A9: e1 is_reflexive_in X by RELAT_2:def 9;
  ex F1 being non empty FinSequence of X st len F1 = n+2 & x,y are_joint_by
  F1,e1,e2 by A2,A5;
  hence contradiction by A6,A9,A8,A7,Th12,NAT_1:11;
end;
