reserve X for BCI-algebra;
reserve n for Nat;
reserve x,y for Element of X;
reserve a,b for Element of AtomSet(X);
reserve m,n for Nat;
reserve i,j for Integer;
reserve X,X9,Y for BCI-algebra,
  H9 for SubAlgebra of X9,
  G for SubAlgebra of X,

  A9 for non empty Subset of X9,
  I for Ideal of X,
  CI,K for closed Ideal of X,
  x,y,a,b for Element of X,
  RI for I-congruence of X,I,
  RK for I-congruence of X,K;
reserve f for BCI-homomorphism of X,X9;
reserve g for BCI-homomorphism of X9,X;
reserve h for BCI-homomorphism of X9,Y;

theorem Th54:
  HK(G,RK) is BCI-algebra
proof
  reconsider IT = HK(G,RK) as non empty BCIStr_0;
A1: IT is being_BCI-4
  proof
    let x,y be Element of IT;
    reconsider x1=x,y1=y as Element of Union(G,RK);
    reconsider x2=x1,y2=y1 as Element of X;
    assume x\y=0.IT & y\x=0.IT;
    then x2\y2 = 0.X & y2\x2 = 0.X by Def12;
    hence thesis by BCIALG_1:def 7;
  end;
A2: now
    let x,y,z be Element of IT;
    reconsider x1=x,y1=y,z1=z as Element of Union(G,RK);
    reconsider x2=x1,y2=y1,z2=z1 as Element of X;
    x2\y2 = x1\y1 & z2\y2=z1\y1 by Def12;
    then
A3: (x2\y2)\(z2\y2) = x1\y1\(z1\y1) by Def12;
    x2\z2=x1\z1 by Def12;
    then ((x2\y2)\(z2\y2))\(x2\z2)=((x1\y1)\(z1\y1))\(x1\z1)by A3,Def12;
    hence ((x\y)\(z\y))\(x\z)=0.IT by BCIALG_1:def 3;
  end;
A4: IT is being_C
  proof
    let x,y,z be Element of IT;
    reconsider x1=x,y1=y,z1=z as Element of Union(G,RK);
    reconsider x2=x1,y2=y1,z2=z1 as Element of X;
    x2\z2=x1\z1 by Def12;
    then
A5: (x2\z2)\y2=(x1\z1)\y1 by Def12;
    x2\y2 = x1\y1 by Def12;
    then (x2\y2)\z2 = (x1\y1)\z1 by Def12;
    then ((x2\y2)\z2)\((x2\z2)\y2)=((x1\y1)\z1)\((x1\z1)\y1) by A5,Def12;
    hence thesis by BCIALG_1:def 4;
  end;
  IT is being_I
  proof
    let x be Element of IT;
    reconsider x1=x as Element of Union(G,RK);
    reconsider x2=x1 as Element of X;
    x2\x2=x1\x1 by Def12;
    hence thesis by BCIALG_1:def 5;
  end;
  hence thesis by A2,A4,A1,BCIALG_1:def 3;
end;
