
theorem Th10:
  for L being RelStr, X being set holds ex_sup_of X,L iff ex_inf_of X,L opp
proof
  let L be RelStr, X be set;
  hereby
    assume ex_sup_of X,L;
    then consider a being Element of L such that
A1: X is_<=_than a and
A2: for b being Element of L st X is_<=_than b holds b >= a and
A3: for c being Element of L st X is_<=_than c & for b being Element
    of L st X is_<=_than b holds b >= c holds c = a;
    thus ex_inf_of X,L opp
    proof
      take a~;
      thus X is_>=_than a~ by A1,Th8;
      hereby
        let b be Element of L opp;
        assume X is_>=_than b;
        then X is_<=_than ~b by Th9;
        hence b <= a~ by Th2,A2;
      end;
      let c be Element of L opp;
      assume X is_>=_than c;
      then
A4:   X is_<=_than ~c by Th9;
      assume
A5:   for b being Element of L opp st X is_>=_than b holds b <= c;
      now
        let b be Element of L;
        assume X is_<=_than b;
        then X is_>=_than b~ by Th8;
        hence b >= ~c by Th2,A5;
      end;
      hence thesis by A3,A4;
    end;
  end;
  assume ex_inf_of X,L opp;
  then consider a being Element of L opp such that
A6: X is_>=_than a and
A7: for b being Element of L opp st X is_>=_than b holds b <= a and
A8: for c being Element of L opp st X is_>=_than c & for b being
  Element of L opp st X is_>=_than b holds b <= c holds c = a;
  take ~a;
  thus X is_<=_than ~a by A6,Th9;
  hereby
    let b be Element of L;
    assume X is_<=_than b;
    then X is_>=_than b~ by Th8;
    hence b >= ~a by Th2,A7;
  end;
  let c be Element of L;
  assume X is_<=_than c;
  then
A9: X is_>=_than c~ by Th8;
  assume
A10: for b being Element of L st X is_<=_than b holds b >= c;
  now
    let b be Element of L opp;
    assume X is_>=_than b;
    then X is_<=_than ~b by Th9;
    hence b <= c~ by Th2,A10;
  end;
  hence thesis by A8,A9;
end;
