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 Th42:
  [x,y] in E & u in Class(E,0.X) implies [x,(y,u)to_power k] in E
proof
  assume that
A1: [x,y] in E and
A2: u in Class(E,0.X);
  defpred P[Nat] means [x,(y,u)to_power $1] in E;
  ex z being object st [z,u] in E & z in {0.X} by A2,RELAT_1:def 13;
  then
A3: [0.X,u] in E by TARSKI:def 1;
A4: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume [x,(y,u) to_power k] in E;
    then [x\0.X,(y,u)to_power k\u] in E by A3,Def9;
    then [x,(y,u)to_power k\u] in E by BCIALG_1:2;
    hence thesis by Th4;
  end;
A5: P[0] by A1,Th1;
  for n holds P[n] from NAT_1:sch 2(A5,A4);
  hence thesis;
end;
