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 X,x,y holds ex i,j,m,n st ((x,x\y) to_power i,y\x) to_power j = (
(y,y\x)to_power m,x\y) to_power n) implies for E,I st I=Class(E,0.X) holds E is
  I-congruence of X,I
proof
  assume
A1: for X,x,y holds ex i,j,m,n st ((x,x\y)to_power i,y\x)to_power j =((y
  ,y\x)to_power m,x\y)to_power n;
  let E,I;
  assume
A2: I=Class(E,0.X);
  now
    let x,y be Element of X;
    x\y in I & y\x in I implies[x,y] in E
    proof
      assume that
A3:   x\y in I and
A4:   y\x in I;
      ex z being object st [z,y\x] in E & z in {0.X} by A2,A4,RELAT_1:def 13;
      then
A5:   [0.X,y\x] in E by TARSKI:def 1;
      ex z being object st [z,x\y] in E & z in {0.X} by A2,A3,RELAT_1:def 13;
      then
A6:   [0.X,x\y] in E by TARSKI:def 1;
      consider i,j,m,n being Nat such that
A7:   ((x,x\y)to_power i,y\x)to_power j =((y,y\x)to_power m,x\y)
      to_power n by A1;
A8:   field E = the carrier of X by EQREL_1:9;
A9:   E is_reflexive_in field E by RELAT_2:def 9;
      then [x,x] in E by A8,RELAT_2:def 1;
      then
A10:  [x,(x,x\y) to_power i] in E by A2,A3,Th42;
A11:  [x,((x,(x\y))to_power i,y\x)to_power j] in E
      proof
        defpred P[Nat] means [x,((x,(x\y))to_power i,y\x)to_power $1] in E;
A12:    for k be Nat st P[k] holds P[k+1]
        proof
          let k be Nat;
          assume [x,((x,(x\y))to_power i,y\x)to_power k] in E;
          then [x\0.X,((x,(x\y))to_power i,y\x)to_power k\(y\x)] in E by A5
,Def9;
          then [x,((x,(x\y))to_power i,y\x)to_power k\(y\x)] in E by BCIALG_1:2
;
          hence thesis by Th4;
        end;
A13:    P[0] by A10,Th1;
        for n holds P[n] from NAT_1:sch 2(A13,A12);
        hence thesis;
      end;
      [y,y] in E by A8,A9,RELAT_2:def 1;
      then
A14:  [y,(y,y\x) to_power m] in E by A2,A4,Th42;
A15:  [y,((y,(y\x))to_power m,x\y)to_power n] in E
      proof
        defpred P[Nat] means [y,((y,(y\x))to_power m,x\y)to_power $1] in E;
A16:    for k be Nat st P[k] holds P[k+1]
        proof
          let k be Nat;
          assume [y,((y,(y\x))to_power m,x\y)to_power k] in E;
          then [y\0.X,((y,(y\x))to_power m,x\y)to_power k\(x\y)] in E by A6
,Def9;
          then [y,((y,(y\x))to_power m,x\y)to_power k\(x\y)] in E by BCIALG_1:2
;
          hence thesis by Th4;
        end;
A17:    P[0] by A14,Th1;
        for n holds P[n] from NAT_1:sch 2(A17,A16);
        hence thesis;
      end;
      E is_symmetric_in field E by RELAT_2:def 11;
      then
      E is_transitive_in field E & [((x,(x\y))to_power i,y\x)to_power j,y
      ] in E by A7,A8,A15,RELAT_2:def 3,def 16;
      hence thesis by A8,A11,RELAT_2:def 8;
    end;
    hence [x,y] in E iff x\y in I & y\x in I by A2,Th40;
  end;
  hence thesis by Def12;
end;
