reserve X for BCI-algebra;
reserve x,y,z,u,a,b for Element of X;
reserve IT for non empty Subset of X;

theorem Th39:
  for a,b being Element of AtomSet(X),x being Element of BranchV(a
  ), y being Element of BranchV(b) holds x\y in BranchV(a\b)
proof
  let a,b be Element of AtomSet(X),x be Element of BranchV(a), y be Element of
  BranchV(b);
  (a\b)\(x\y)=(((a\b)`)`)\(x\y) by Th29;
  then (a\b)\(x\y)=(x\y)`\(a\b)` by Th7;
  then (a\b)\(x\y)=x`\y`\(a\b)` by Th9;
  then (a\b)\(x\y)=((x`)\((a\b)`))\y` by Th7;
  then (a\b)\(x\y)=((((a\b)`)`)\x)\(y`) by Th7;
  then (a\b)\(x\y)=((a\b)\x)\(y`) by Th29;
  then (a\b)\(x\y)=((a\x)\b)\(y`) by Th7;
  then (a\b)\(x\y)=b`\y` by Lm2;
  then (a\b)\(x\y)=(b\y)` by Th9;
  then (a\b)\(x\y)=(0.X)` by Lm2;
  then (a\b)\(x\y)=0.X by Def5;
  then a\b <= x\y;
  hence thesis;
end;
