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
  0.X is maximal implies for a holds a is minimal
proof
  assume
A1: 0.X is maximal;
  let a;
  now
    let x;
    assume x<=a;
    then
A2: x\a=0.X;
    then (a\x)`=0.X by BCIALG_1:6;
    then 0.X <= a\x;
    then 0.X = a\x by A1;
    hence x=a by A2,BCIALG_1:def 7;
  end;
  hence thesis by Lm1;
end;
