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
  X is p-Semisimple iff for x,y,z,u holds (x\u)\(z\y) = (y\u)\(z\x)
proof
  thus X is p-Semisimple implies for x,y,z,u holds (x\u)\(z\y) = (y\u)\(z\x)
  by Lm9;
  thus (for x,y,z,u holds (x\u)\(z\y) = (y\u)\(z\x)) implies X is p-Semisimple
  proof
    assume
A1: for x,y,z,u holds (x\u)\(z\y) = (y\u)\(z\x);
    for x,y,z holds x\(z\y) = y\(z\x)
    proof
      let x,y,z;
      (x\0.X)\(z\y) = (y\0.X)\(z\x) by A1;
      then x\(z\y)= (y\0.X)\(z\x) by Th2;
      hence thesis by Th2;
    end;
    hence thesis by Th57;
  end;
end;
