reserve X for BCI-algebra;
reserve I for Ideal of X;
reserve a,x,y,z,u for Element of X;
reserve f,f9,g for sequence of  the carrier of X;
reserve j,i,k,n,m for Nat;

theorem
  x` is minimal iff for y holds y<=x implies x` = y`
proof
  thus x` is minimal implies for y holds y<=x implies x` = y`
  proof
    assume
A1: x` is minimal;
    let y;
    assume y<=x;
    then y\x=0.X;
    then (y\x)`=0.X by BCIALG_1:def 5;
    then y`\x`=0.X by BCIALG_1:9;
    then y`<=x`;
    hence thesis by A1;
  end;
  thus (for y holds y<=x implies x` = y`)implies x` is minimal
  proof
    assume
A2: for y holds y<=x implies x` = y`;
    now
      let xx be Element of X;
      assume xx<=x`;
      then
A3:   xx\x`=0.X;
      then (xx\x`)`=0.X by BCIALG_1:def 5;
      then xx`\((x`)`)=0.X by BCIALG_1:9;
      then (xx`\((x`)`))`=0.X by BCIALG_1:def 5;
      then ((xx`)`)\(((x`)`)`)=0.X by BCIALG_1:9;
      then ((xx`)`)\x`=0.X by BCIALG_1:8;
      then (xx`\x)`=0.X by BCIALG_1:9;
      then (x`\xx)`=0.X by BCIALG_1:7;
      then (xx\xx)\(x`\xx)=0.X by BCIALG_1:def 5;
      then (xx\x`)`=0.X by BCIALG_1:def 3;
      then xx`\((x`)`)=0.X by BCIALG_1:9;
      then xx`\x\(((x`)`)\x)=0.X by BCIALG_1:4;
      then xx`\x\0.X=0.X by BCIALG_1:1;
      then xx`\x=0.X by BCIALG_1:2;
      then xx`<=x;
      then (xx`)`=x` by A2;
      then 0.X = x`\xx by BCIALG_1:1;
      hence xx=x` by A3,BCIALG_1:def 7;
    end;
    hence thesis by Lm1;
  end;
end;
