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
  a is minimal iff for x holds a\x=x`\a`
proof
  thus a is minimal implies for x holds a\x=x`\a`
  proof
    assume
A1: a is minimal;
    let x;
    x\(x\a)\a=0.X by BCIALG_1:1;
    then x\(x\a)<=a;
    then
A2: (a\x)\(x`\a`)=((x\(x\a))\x)\(x`\a`)by A1
      .=((x\x)\(x\a))\(x`\a`) by BCIALG_1:7
      .=((x\a)`)\(x`\a`) by BCIALG_1:def 5
      .=((x\a)`)\((x\a)`) by BCIALG_1:9
      .=0.X by BCIALG_1:def 5;
    (x`\a`)\(a\x)=0.X by BCIALG_1:1;
    hence thesis by A2,BCIALG_1:def 7;
  end;
  thus (for x holds a\x=x`\a`) implies a is minimal
  proof
    assume
A3: for x holds a\x=x`\a`;
    now
      let x;
      assume x<=a;
      then
A4:   x\a=0.X;
      a\x=x`\a` by A3;
      then a\x=(0.X)` by A4,BCIALG_1:9;
      then a\x=0.X by BCIALG_1:def 5;
      hence a=x by A4,BCIALG_1:def 7;
    end;
    hence thesis by Lm1;
  end;
end;
