reserve x,y,z,r,s for ExtReal;
reserve A,B for ext-real-membered set;
reserve A,B for ext-real-membered set;

theorem
  for A being left_end non right_end interval ext-real-membered set
  holds A = [.min A, sup A.[
proof
  let A be left_end non right_end interval ext-real-membered set;
  let x;
  defpred P[ExtReal] means $1 in A & $1 > x;
  thus x in A implies x in [.min A, sup A.[
  proof
A1: not sup A in A by Def6;
    assume
A2: x in A;
    then
A3: min A <= x by Th3;
    x <= sup A by A2,Th4;
    then x < sup A by A2,A1,XXREAL_0:1;
    hence thesis by A3,XXREAL_1:3;
  end;
  assume
A4: x in [.min A, sup A.[;
  per cases;
  suppose
    not ex r st P[r];
    then x is UpperBound of A by Def1;
    then sup A <= x by Def3;
    hence thesis by A4,XXREAL_1:3;
  end;
  suppose
    ex r st P[r];
    then consider r such that
A5: r in A and
A6: r > x;
    inf A <= x by A4,XXREAL_1:3;
    then
A7: x in [.inf A,r.] by A6,XXREAL_1:1;
    min A in A by Def5;
    then [.inf A,r.] c= A by A5,Def12;
    hence thesis by A7;
  end;
end;
