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 Th37:
  RI is Congruence of X
proof
  field RI = the carrier of X by EQREL_1:9;
  then
A1: RI is_transitive_in the carrier of X by RELAT_2:def 16;
  now
    let x,y,u,v be Element of X;
    assume that
A2: [x,y] in RI and
A3: [u,v]in RI;
    (y\u\(y\v))\(v\u)=0.X by BCIALG_1:1;
    then
A4: (y\u\(y\v))\(v\u) in I by BCIALG_1:def 18;
    v\u in I by A3,Def12;
    then
A5: y\u\(y\v) in I by A4,BCIALG_1:def 18;
    (y\v\(y\u))\(u\v)=0.X by BCIALG_1:1;
    then
A6: (y\v\(y\u))\(u\v) in I by BCIALG_1:def 18;
    u\v in I by A3,Def12;
    then y\v\(y\u) in I by A6,BCIALG_1:def 18;
    then
A7: [y\u,y\v] in RI by A5,Def12;
    (x\u\(y\u))\(x\y)=0.X by BCIALG_1:def 3;
    then
A8: (x\u\(y\u))\(x\y) in I by BCIALG_1:def 18;
    (y\u\(x\u))\(y\x)=0.X by BCIALG_1:def 3;
    then
A9: (y\u\(x\u))\(y\x) in I by BCIALG_1:def 18;
    y\x in I by A2,Def12;
    then
A10: y\u\(x\u) in I by A9,BCIALG_1:def 18;
    x\y in I by A2,Def12;
    then x\u\(y\u) in I by A8,BCIALG_1:def 18;
    then [x\u,y\u] in RI by A10,Def12;
    hence [x\u,y\v] in RI by A1,A7,RELAT_2:def 8;
  end;
  hence thesis by Def9;
end;
