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 Th57:
  X is p-Semisimple iff for x,y,z holds x\(z\y) = y\(z\x)
proof
  thus X is p-Semisimple implies for x,y,z holds x\(z\y) = y\(z\x)
  proof
    assume
A1: X is p-Semisimple;
    let x,y,z;
    y\(z\x) =(z\(z\y))\(z\x) by A1;
    then
A2: (y\(z\x))\(x\(z\y))=0.X by Th1;
    x\(z\y) =(z\(z\x))\(z\y)by A1;
    then (x\(z\y))\(y\(z\x))=0.X by Th1;
    hence thesis by A2,Def7;
  end;
  assume
A3: for x,y,z holds x\(z\y) = y\(z\x);
  for x holds x`` = x
  proof
    let x;
    x`` = x\(0.X)` by A3
      .=x\0.X by Def5;
    hence thesis by Th2;
  end;
  hence thesis by Th54;
end;
