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 Th40:
  [x,y] in E implies x\y in Class(E,0.X) & y\x in Class(E,0.X)
proof
  assume
A1: [x,y] in E;
A2: field E = the carrier of X by EQREL_1:9;
  then
A3: E is_reflexive_in the carrier of X by RELAT_2:def 9;
  then
A4: [y,y] in E by RELAT_2:def 1;
  E is_symmetric_in the carrier of X by A2,RELAT_2:def 11;
  then [y,x] in E by A1,RELAT_2:def 3;
  then [y\y,x\y] in E by A4,Def9;
  then
A5: [0.X,x\y] in E by BCIALG_1:def 5;
  [x,x] in E by A3,RELAT_2:def 1;
  then [x\x,y\x] in E by A1,Def9;
  then 0.X in {0.X} & [0.X,y\x] in E by BCIALG_1:def 5,TARSKI:def 1;
  hence thesis by A5,RELAT_1:def 13;
end;
