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 Th38:
  for a being Element of AtomSet(X),x being Element of BCK-part(X) holds a\x =a
proof
  let a be Element of AtomSet(X),x be Element of BCK-part(X);
  a\0.X in {x1 where x1 is Element of X:x1 is atom} by Th33;
  then
A1: ex x1 being Element of X st a\0.X=x1 & x1 is atom;
  (a\x)\(a\0.X)=(a\(a\0.X))\x by Th7;
  then (a\x)\(a\0.X)=(a\a)\x by Th2;
  then
A2: (a\x)\(a\0.X)=x` by Def5;
  x in {x2 where x2 is Element of X:0.X<=x2};
  then ex x2 being Element of X st x=x2 & 0.X<=x2;
  then (a\x)\(a\0.X)=0.X by A2;
  then a\x=a\0.X by A1;
  hence thesis by Th2;
end;
