reserve X for non empty set;

theorem Th10:
  for A being set, a,b being Element of EqRelLATT A for E1,E2
being Equivalence_Relation of A st a = E1 & b = E2 holds a "\/" b = E1 "\/" E2
proof
  let A be set, a,b be Element of EqRelLATT A, E1,E2 be Equivalence_Relation
  of A;
  assume
A1: a = E1 & b = E2;
  reconsider y = b as Element of LattPOSet EqRelLatt A;
  reconsider x = a as Element of LattPOSet EqRelLatt A;
  reconsider x as Element of EqRelLatt A;
  reconsider y as Element of EqRelLatt A;
  %(x%) = x% & %(y%) = y%;
  hence a "\/" b = x "\/" y by Th9
    .= (the L_join of EqRelLatt A).(x,y) by LATTICES:def 1
    .= E1 "\/" E2 by A1,MSUALG_5:def 2;
end;
