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 Th76:
  X is alternative implies x` = x & x\(x\y) = y & (x\y)\y = x
proof
  assume
A1: X is alternative;
  then x\(x\x) = (x\x)\x;
  then x\0.X = (x\x)\x by Def5;
  then x\0.X = x` by Def5;
  hence x` = x by Th2;
  y\(y\y) = (y\y)\y by A1;
  then y\0.X = (y\y)\y by Def5;
  then y\0.X = y` by Def5;
  then
A2: y = y` by Th2;
  x\(x\y) = (x\x)\y by A1;
  hence x\(x\y) = y by A2,Def5;
  (x\y)\y=x\(y\y) by A1;
  then (x\y)\y=x\0.X by Def5;
  hence thesis by Th2;
end;
