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
  (for LC holds LC is I-congruence of X,I) implies E = RI
proof
  assume
A1: for LC holds LC is I-congruence of X,I;
  let x1,y1 be object;
  E is L-congruence of X by Th36;
  then
A2: E is I-congruence of X,I by A1;
  thus [x1,y1] in E implies [x1,y1] in RI
  proof
    assume
A3: [x1,y1] in E;
    then consider x,y being object such that
A4: [x1,y1]=[x,y] and
A5: x in the carrier of X & y in the carrier of X by RELSET_1:2;
    reconsider x,y as Element of X by A5;
    y\x in Class(E,0.X) by A3,A4,Th40;
    then ex z being object st [z,y\x] in E & z in {0.X} by RELAT_1:def 13;
    then [0.X,y\x] in E by TARSKI:def 1;
    then y\x\0.X in I by A2,Def12;
    then
A6: y\x in I by BCIALG_1:2;
    x\y in Class(E,0.X) by A3,A4,Th40;
    then ex z being object st [z,x\y] in E & z in {0.X} by RELAT_1:def 13;
    then [0.X,x\y] in E by TARSKI:def 1;
    then x\y\0.X in I by A2,Def12;
    then x\y in I by BCIALG_1:2;
    hence thesis by A4,A6,Def12;
  end;
  thus [x1,y1] in RI implies [x1,y1] in E
  proof
    assume
A7: [x1,y1] in RI;
    then consider x,y being object such that
A8: [x1,y1]=[x,y] and
A9: x in the carrier of X & y in the carrier of X by RELSET_1:2;
    reconsider x,y as Element of X by A9;
    x\y in I & y\x in I by A7,A8,Def12;
    hence thesis by A2,A8,Def12;
  end;
end;
