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;
reserve E for Congruence of X;
reserve RI for I-congruence of X,I;
reserve W1,W2 for Element of Class E;

theorem Th51:
  X./.RI is BCI-algebra
proof
  reconsider IT = X./.RI as non empty BCIStr_0;
A1: now
    let w1,w2,w3 be Element of IT;
    w1 in the carrier of IT;
    then consider x1 being object such that
A2: x1 in the carrier of X and
A3: w1 = Class(RI,x1) by EQREL_1:def 3;
    w3 in the carrier of IT;
    then consider z1 being object such that
A4: z1 in the carrier of X and
A5: w3 = Class(RI,z1) by EQREL_1:def 3;
    w2 in the carrier of IT;
    then consider y1 being object such that
A6: y1 in the carrier of X and
A7: w2 = Class(RI,y1) by EQREL_1:def 3;
    reconsider x1,y1,z1 as Element of X by A2,A6,A4;
A8: w3\w2=Class(RI,z1\y1) by A7,A5,Def17;
    w1\w2=Class(RI,x1\y1) by A3,A7,Def17;
    then
A9: (w1\w2)\(w3\w2)=Class(RI,(x1\y1)\(z1\y1)) by A8,Def17;
    w1\w3=Class(RI,x1\z1) by A3,A5,Def17;
    then (w1\w2)\(w3\w2)\(w1\w3)=Class(RI,(x1\y1)\(z1\y1)\(x1\z1)) by A9,Def17;
    hence ((w1\w2)\(w3\w2))\(w1\w3)=0.IT by BCIALG_1:def 3;
  end;
A10: now
    let w1,w2,w3 be Element of IT;
    w1 in the carrier of IT;
    then consider x1 being object such that
A11: x1 in the carrier of X and
A12: w1 = Class(RI,x1) by EQREL_1:def 3;
    w2 in the carrier of IT;
    then consider y1 being object such that
A13: y1 in the carrier of X and
A14: w2 = Class(RI,y1) by EQREL_1:def 3;
    w3 in the carrier of IT;
    then consider z1 being object such that
A15: z1 in the carrier of X and
A16: w3 = Class(RI,z1) by EQREL_1:def 3;
    reconsider x1,y1,z1 as Element of X by A11,A13,A15;
    w1\w3=Class(RI,x1\z1) by A12,A16,Def17;
    then
A17: (w1\w3)\w2=Class(RI,x1\z1\y1) by A14,Def17;
    w1\w2=Class(RI,x1\y1) by A12,A14,Def17;
    then (w1\w2)\w3=Class(RI,(x1\y1)\z1) by A16,Def17;
    then ((w1\w2)\w3)\((w1\w3)\w2)=Class(RI,(x1\y1)\z1\(x1\z1\y1)) by A17,Def17
;
    hence ((w1\w2)\w3)\((w1\w3)\w2)=0.IT by BCIALG_1:def 4;
  end;
A18: now
    let w1,w2 be Element of IT;
    assume that
A19: w1\w2=0.IT and
A20: w2\w1=0.IT;
    w1 in the carrier of IT;
    then consider x1 being object such that
A21: x1 in the carrier of X and
A22: w1 = Class(RI,x1) by EQREL_1:def 3;
    w2 in the carrier of IT;
    then consider y1 being object such that
A23: y1 in the carrier of X and
A24: w2 = Class(RI,y1) by EQREL_1:def 3;
    reconsider x1,y1 as Element of X by A21,A23;
    w2\w1=Class(RI,y1\x1) by A22,A24,Def17;
    then 0.X in Class(RI,y1\x1) by A20,EQREL_1:23;
    then [y1\x1,0.X] in RI by EQREL_1:18;
    then y1\x1\0.X in I by Def12;
    then
A25: y1\x1 in I by BCIALG_1:2;
    w1\w2=Class(RI,x1\y1) by A22,A24,Def17;
    then 0.X in Class(RI,x1\y1) by A19,EQREL_1:23;
    then [x1\y1,0.X] in RI by EQREL_1:18;
    then x1\y1\0.X in I by Def12;
    then x1\y1 in I by BCIALG_1:2;
    then [x1,y1] in RI by A25,Def12;
    hence w1=w2 by A22,A24,EQREL_1:35;
  end;
  now
    let w1 be Element of IT;
    w1 in the carrier of IT;
    then consider x1 being object such that
A26: x1 in the carrier of X and
A27: w1 = Class(RI,x1) by EQREL_1:def 3;
    reconsider x1 as Element of X by A26;
    w1\w1=Class(RI,x1\x1) by A27,Def17;
    hence w1\w1 = 0.IT by BCIALG_1:def 5;
  end;
  hence thesis by A1,A10,A18,BCIALG_1:def 3,def 4,def 5,def 7;
end;
