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 Th63:
  X is p-Semisimple iff for x,y holds x\y`=y\x`
proof
  thus X is p-Semisimple implies for x,y holds x\y`=y\x` by Th57;
  assume
A1: for x,y holds x\y`=y\x`;
  now
    let x;
    x\(0.X)`=(x`)` by A1;
    then x\0.X=(x`)` by Th2;
    hence x=x`` by Th2;
  end;
  hence thesis by Th54;
end;
