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 Th6:
  for Y be set for X be Subset of EqRelLatt Y holds union X is Relation of Y
proof
  let Y be set;
  let X be Subset of EqRelLatt Y;
  now
    let x be object;
    assume x in union X;
    then consider X9 be set such that
A1: x in X9 and
A2: X9 in X by TARSKI:def 4;
    X9 is Equivalence_Relation of Y by A2,MSUALG_5:21;
    hence x in [:Y,Y:] by A1;
  end;
  hence thesis by TARSKI:def 3;
end;
