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
  I is closed iff I = Class(RI,0.X)
proof
  thus I is closed implies I = Class(RI,0.X)
  proof
    assume
A1: I is closed;
    thus I c= Class(RI,0.X)
    proof
      let x be object;
      assume
A2:   x in I;
      then reconsider x as Element of I;
A3:   x\0.X in I by A2,BCIALG_1:2;
      x` in I by A1;
      then 0.X in {0.X} & [0.X,x] in RI by A3,Def12,TARSKI:def 1;
      hence thesis by RELAT_1:def 13;
    end;
    thus thesis by Th38;
  end;
  assume
A4: I = Class(RI,0.X);
  now
    let x be Element of I;
    ex y being object st [y,x] in RI & y in {0.X} by A4,RELAT_1:def 13;
    then [0.X,x] in RI by TARSKI:def 1;
    hence x` in I by Def12;
  end;
  hence thesis;
end;
