
theorem Th19:
  for L being RelStr for X being Subset of L, Y being Subset of L
  opp st X = Y holds downarrow X = uparrow Y & uparrow X = downarrow Y
proof
  let L be RelStr;
  let X be Subset of L, Y be Subset of L opp such that
A1: X = Y;
  thus downarrow X c= uparrow Y
  proof
    let x be object;
    assume
A2: x in downarrow X;
    then reconsider x as Element of L;
    consider y being Element of L such that
A3: y >= x and
A4: y in X by A2,WAYBEL_0:def 15;
    y~ <= x~ by A3,LATTICE3:9;
    hence thesis by A1,A4,WAYBEL_0:def 16;
  end;
  thus uparrow Y c= downarrow X
  proof
    let x be object;
    assume
A5: x in uparrow Y;
    then reconsider x as Element of L opp;
    consider y being Element of L opp such that
A6: y <= x and
A7: y in Y by A5,WAYBEL_0:def 16;
    ~y >= ~x by A6,Th1;
    hence thesis by A1,A7,WAYBEL_0:def 15;
  end;
  thus uparrow X c= downarrow Y
  proof
    let x be object;
    assume
A8: x in uparrow X;
    then reconsider x as Element of L;
    consider y being Element of L such that
A9: y <= x and
A10: y in X by A8,WAYBEL_0:def 16;
    y~ >= x~ by A9,LATTICE3:9;
    hence thesis by A1,A10,WAYBEL_0:def 15;
  end;
  let x be object;
  assume
A11: x in downarrow Y;
  then reconsider x as Element of L opp;
  consider y being Element of L opp such that
A12: y >= x and
A13: y in Y by A11,WAYBEL_0:def 15;
  ~y <= ~x by A12,Th1;
  hence thesis by A1,A13,WAYBEL_0:def 16;
end;
