reserve X for BCI-algebra;
reserve I for Ideal of X;
reserve a,x,y,z,u for Element of X;
reserve f,f9,g for sequence of  the carrier of X;
reserve j,i,k,n,m for Nat;
reserve R for Equivalence_Relation of X;
reserve RI for I-congruence of X,I;
reserve E for Congruence of X;
reserve RC for R-congruence of X;
reserve LC for L-congruence of X;

theorem
  ConSet(X) = LConSet(X)/\RConSet(X)
proof
  ConSet(X) c= LConSet(X) & ConSet(X) c= RConSet(X) by Th45,Th46;
  hence ConSet(X) c= LConSet(X)/\RConSet(X) by XBOOLE_1:19;
  thus LConSet(X)/\RConSet(X) c= ConSet(X)
  proof
    let x be object;
    assume
A1: x in LConSet(X)/\RConSet(X);
    then x in RConSet(X) by XBOOLE_0:def 4;
    then
A2: ex R being R-congruence of X st x=R;
    x in LConSet(X) by A1,XBOOLE_0:def 4;
    then ex L being L-congruence of X st x=L;
    then x is Congruence of X by A2,Th36;
    hence thesis;
  end;
end;
