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 Th6:
  ( EqR1 (\/) EqR2 c= EqR3 & for EqR be Equivalence_Relation of M st
  EqR1 (\/) EqR2 c= EqR holds EqR3 c= EqR ) implies EqR3 = EqR1 "\/" EqR2
proof
  assume that
A1: EqR1 (\/) EqR2 c= EqR3 and
A2: for EqR be Equivalence_Relation of M st EqR1 (\/) EqR2 c= EqR holds
  EqR3 c= EqR;
A3: EqR1 "\/" EqR2 c= EqR3 by A1,Th5;
  EqR3 c= EqR1 "\/" EqR2 by A2,Th4;
  hence thesis by A3,PBOOLE:146;
end;
