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
  (ex x st x is greatest) implies for a holds a is positive
proof
  given x such that
A1: x is greatest;
  let a;
  a<=x by A1;
  then a\x=0.X;
  then
A2: (a\x)`=0.X by BCIALG_1:def 5;
  0.X<=x by A1;
  then x`=0.X;
  then a`\0.X=0.X by A2,BCIALG_1:9;
  then a`=0.X by BCIALG_1:2;
  then 0.X<=a;
  hence thesis;
end;
