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 Th36:
  R is Congruence of X iff R is R-congruence of X&R is L-congruence of X
proof
A1: field R = the carrier of X by EQREL_1:9;
  then
A2: R is_reflexive_in the carrier of X by RELAT_2:def 9;
  thus R is Congruence of X implies R is R-congruence of X&R is L-congruence
  of X
  proof
    assume
A3: R is Congruence of X;
    thus R is R-congruence of X
    proof
      let x,y be Element of X;
      assume
A4:   [x,y] in R;
      let u be Element of X;
      [u,u] in R by A2,RELAT_2:def 1;
      hence thesis by A3,A4,Def9;
    end;
    let x,y be Element of X;
    assume
A5: [x,y] in R;
    let u be Element of X;
    [u,u] in R by A2,RELAT_2:def 1;
    hence thesis by A3,A5,Def9;
  end;
  assume
A6: R is R-congruence of X & R is L-congruence of X;
A7: R is_transitive_in the carrier of X by A1,RELAT_2:def 16;
  now
    let x,y,u,v be Element of X;
    assume [x,y] in R & [u,v]in R;
    then [x\u,y\u] in R & [y\u,y\v] in R by A6,Def10,Def11;
    hence [x\u,y\v] in R by A7,RELAT_2:def 8;
  end;
  hence thesis by Def9;
end;
