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;

theorem Th34:
  for X being BCI-algebra for Y being SubAlgebra of X holds for x,
y being Element of X,x9,y9 being Element of Y st x = x9 & y = y9 holds x\y = x9
  \y9
proof
  let X be BCI-algebra;
  let Y be SubAlgebra of X;
  let x,y be Element of X,x9,y9 be Element of Y such that
A1: x = x9 & y = y9;
A2: [x9,y9] in [:the carrier of Y,the carrier of Y:] by ZFMISC_1:87;
  x9\y9 = ((the InternalDiff of X)||the carrier of Y).(x9,y9) by
BCIALG_1:def 10
    .= x\y by A1,A2,FUNCT_1:49;
  hence thesis;
end;
