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 Th30:
  a is minimal iff ex x st a=x`
proof
  thus a is minimal implies ex x st a=x`
  proof
    assume
A1: a is minimal;
    take x=a`;
    x`\a=0.X by BCIALG_1:1;
    then x` <= a;
    hence thesis by A1;
  end;
  given x such that
A2: a=x`;
  now
    let y;
    assume y<=a;
    then
A3: y\a=0.X;
    then a\y=y\x`\y\x by A2,BCIALG_1:7;
    then a\y=y\y\x`\x by BCIALG_1:7;
    then a\y=(x`)`\x by BCIALG_1:def 5;
    then a\y=0.X by BCIALG_1:1;
    hence a=y by A3,BCIALG_1:def 7;
  end;
  hence thesis by Lm1;
end;
