reserve I,X,x,d,i for set;
reserve M for ManySortedSet of I;
reserve EqR1,EqR2 for Equivalence_Relation of X;
reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve EqR,EqR1,EqR2,EqR3,EqR4 for Equivalence_Relation of M;

theorem Th5:
  for EqR be Equivalence_Relation of M st EqR1 (\/) EqR2 c= EqR holds
  EqR1 "\/" EqR2 c= EqR
proof
  consider EqR3 being ManySortedRelation of M such that
A1: EqR3 = EqR1 (\/) EqR2 and
A2: EqR1 "\/" EqR2 = EqCl EqR3 by Def4;
  let EqR be Equivalence_Relation of M such that
A3: EqR1 (\/) EqR2 c= EqR;
  now
    let i be object;
    assume
A4: i in I;
    then reconsider i9 = i as Element of I;
    EqR.i9 is Relation of M.i;
    then reconsider E = EqR.i as Equivalence_Relation of M.i by MSUALG_4:def 2;
    EqR3.i c= E by A3,A1,A4,PBOOLE:def 2;
    then EqCl(EqR3.i9) c= E by Def1;
    hence (EqR1 "\/" EqR2).i c= EqR.i by A2,Def3;
  end;
  hence thesis by PBOOLE:def 2;
end;
