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;
