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;
