
theorem Th14:
  for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2
for X being set holds (ex_sup_of X,L1 implies ex_sup_of X,L2) & (ex_inf_of X,L1
  implies ex_inf_of X,L2)
proof
  let L1,L2 be RelStr such that
A1: the RelStr of L1 = the RelStr of L2;
  let X be set;
  thus ex_sup_of X,L1 implies ex_sup_of X,L2
  proof
    given a being Element of L1 such that
A2: X is_<=_than a and
A3: for b being Element of L1 st X is_<=_than b holds b >= a and
A4: for c being Element of L1 st X is_<=_than c & for b being Element
    of L1 st X is_<=_than b holds b >= c holds c = a;
    reconsider a9 = a as Element of L2 by A1;
    take a9;
    thus X is_<=_than a9 by A1,A2,Th2;
    hereby
      let b9 be Element of L2;
      reconsider b = b9 as Element of L1 by A1;
      assume X is_<=_than b9;
      then b >= a by A1,A3,Th2;
      hence b9 >= a9 by A1;
    end;
    let c9 be Element of L2;
    reconsider c = c9 as Element of L1 by A1;
    assume X is_<=_than c9;
    then
A5: X is_<=_than c by A1,Th2;
    assume
A6: for b9 being Element of L2 st X is_<=_than b9 holds b9 >= c9;
    now
      let b be Element of L1;
      reconsider b9 = b as Element of L2 by A1;
      assume X is_<=_than b;
      then b9 >= c9 by A1,A6,Th2;
      hence b >= c by A1;
    end;
    hence thesis by A4,A5;
  end;
  given a being Element of L1 such that
A7: X is_>=_than a and
A8: for b being Element of L1 st X is_>=_than b holds b <= a and
A9: for c being Element of L1 st X is_>=_than c & for b being Element
  of L1 st X is_>=_than b holds b <= c holds c = a;
  reconsider a9 = a as Element of L2 by A1;
  take a9;
  thus X is_>=_than a9 by A1,A7,Th2;
  hereby
    let b9 be Element of L2;
    reconsider b = b9 as Element of L1 by A1;
    assume X is_>=_than b9;
    then b <= a by A1,A8,Th2;
    hence b9 <= a9 by A1;
  end;
  let c9 be Element of L2;
  reconsider c = c9 as Element of L1 by A1;
  assume
A10: X is_>=_than c9;
  assume
A11: for b9 being Element of L2 st X is_>=_than b9 holds b9 <= c9;
A12: now
    let b be Element of L1;
    reconsider b9 = b as Element of L2 by A1;
    assume X is_>=_than b;
    then b9 <= c9 by A1,A11,Th2;
    hence b <= c by A1;
  end;
  X is_>=_than c by A1,A10,Th2;
  hence thesis by A9,A12;
end;
