reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;
reserve I for I_Lattice,
  i,j,k for Element of I;
reserve B for B_Lattice,
  FB,HB for Filter of B;
reserve I for I_Lattice,
  i,j,k for Element of I,
  DI for non empty Subset of I,
  FI for Filter of I;
reserve F1,F2 for Filter of I;
reserve a,b,c for Element of B;
reserve o1,o2 for BinOp of F;

theorem Th63:
  equivalence_wrt F is Relation of the carrier of L
proof
  equivalence_wrt F c= [:the carrier of L, the carrier of L:]
  proof
    let y,z be object;
    assume [y,z] in equivalence_wrt F;
    then
A1: y in field equivalence_wrt F & z in field equivalence_wrt F by RELAT_1:15;
    field equivalence_wrt F c= the carrier of L by Def11;
    hence thesis by A1,ZFMISC_1:87;
  end;
  hence thesis;
end;
