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 Th8:
  for Y be set for X be Subset of EqRelLatt Y for R be Relation of
  Y st R = union X holds "\/" X = EqCl R
proof
  let Y be set;
  let X be Subset of EqRelLatt Y;
  let R be Relation of Y;
  reconsider X1 = "\/" X as Equivalence_Relation of Y by MSUALG_5:21;
  assume
A1: R = union X;
A2: now
    let EqR be Equivalence_Relation of Y;
    reconsider EqR1 = EqR as Element of EqRelLatt Y by MSUALG_5:21;
    assume
A3: R c= EqR;
    now
      let q be Element of EqRelLatt Y;
      reconsider q1 = q as Equivalence_Relation of Y by MSUALG_5:21;
      assume
A4:   q in X;
      now
        let x be object;
        assume x in q1;
        then x in union X by A4,TARSKI:def 4;
        hence x in EqR by A1,A3;
      end;
      then q1 c= EqR;
      hence q [= EqR1 by Th2;
    end;
    then X is_less_than EqR1 by LATTICE3:def 17;
    then "\/" X [= EqR1 by LATTICE3:def 21;
    hence X1 c= EqR by Th2;
  end;
  R c= "\/" X by A1,Th7;
  hence thesis by A2,MSUALG_5:def 1;
end;
