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