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 Th47:
  X is associative iff for x being Element of X holds x`=x
proof
  thus X is associative implies for x being Element of X holds x`=x
  proof
    assume
A1: X is associative;
    let x be Element of X;
A2: x\x`=(x\0.X)\x by A1
      .=x\x by Th2
      .=0.X by Def5;
    x`\x=(x\x)` by A1
      .=(0.X)` by Def5
      .=0.X by Def5;
    hence thesis by A2,Def7;
  end;
  assume for x being Element of X holds x`=x;
  then for x,y holds x\y=y\x by Lm7;
  hence thesis by Lm6;
end;
