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
  for a being Element of AtomSet(X),b being Element of AtomSet(X9)st b=f
  .a holds f.:BranchV(a) c= BranchV(b)
proof
  let a be Element of AtomSet(X),b be Element of AtomSet(X9) such that
A1: b=f.a;
  let y be object;
  assume y in f.:BranchV(a);
  then consider x being object such that
  x in dom f and
A2: x in BranchV(a) and
A3: y = f.x by FUNCT_1:def 6;
A4: ex x1 being Element of X st x=x1 & a<=x1 by A2;
  reconsider x as Element of X by A2;
  f.a\f.x=f.(a\x) by Def6;
  then f.a\f.x=f.(0.X) by A4;
  then f.a\f.x=0.X9 by Th35;
  then b <= f.x by A1;
  hence thesis by A3;
end;
