reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve Y,x,y,y1,i,j for set;
reserve k for Element of NAT;
reserve p for FinSequence;
reserve S for non void non empty ManySortedSign;
reserve A for non-empty MSAlgebra over S;

theorem Th7:
  for Y be set for X be Subset of EqRelLatt Y holds union X c= "\/" X
proof
  let Y be set;
  let X be Subset of EqRelLatt Y;
  reconsider X9 = "\/" X as Equivalence_Relation of Y by MSUALG_5:21;
  let x be object;
  assume x in union X;
  then consider X1 be set such that
A1: x in X1 and
A2: X1 in X by TARSKI:def 4;
  reconsider X1 as Element of EqRelLatt Y by A2;
  reconsider X2 = X1 as Equivalence_Relation of Y by MSUALG_5:21;
  X is_less_than "\/" X by LATTICE3:def 21;
  then X1 [= "\/" X by A2,LATTICE3:def 17;
  then X2 c= X9 by Th2;
  hence thesis by A1;
end;
