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