
theorem
  for L being non empty RelStr for X being Subset of L, Y being Subset
  of L opp st X = Y holds fininfs X = finsups Y & finsups X = fininfs Y
proof
  let L be non empty RelStr;
  let X be Subset of L, Y be Subset of L opp such that
A1: X = Y;
  thus fininfs X c= finsups Y
  proof
    let x be object;
    assume x in fininfs X;
    then consider Z being finite Subset of X such that
A2: x = "/\"(Z,L) & ex_inf_of Z,L;
    x = "\/"(Z,L opp) & ex_sup_of Z, L opp by A2,Th11,Th13;
    hence thesis by A1;
  end;
  thus finsups Y c= fininfs X
  proof
    let x be object;
    assume x in finsups Y;
    then consider Z being finite Subset of Y such that
A3: x = "\/"(Z,L opp) & ex_sup_of Z,L opp;
    x = "/\"(Z,L) & ex_inf_of Z, L by A3,Th11,Th13;
    hence thesis by A1;
  end;
  thus finsups X c= fininfs Y
  proof
    let x be object;
    assume x in finsups X;
    then consider Z being finite Subset of X such that
A4: x = "\/"(Z,L) & ex_sup_of Z,L;
    x = "/\"(Z,L opp) & ex_inf_of Z, L opp by A4,Th10,Th12;
    hence thesis by A1;
  end;
  let x be object;
  assume x in fininfs Y;
  then consider Z being finite Subset of Y such that
A5: x = "/\"(Z,L opp) & ex_inf_of Z,L opp;
  x = "\/"(Z,L) & ex_sup_of Z, L by A5,Th10,Th12;
  hence thesis by A1;
end;
