
theorem Th13:
  for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2
  for X being Subset of L1 for Y being Subset of L2
  st X = Y holds downarrow X = downarrow Y & uparrow X = uparrow Y
proof
  let L1,L2 be RelStr such that
A1: the RelStr of L1 = the RelStr of L2;
  let X be Subset of L1;
  let Y be Subset of L2 such that
A2: X = Y;
  thus downarrow X c= downarrow Y
  proof
    let x be object;
    assume
A3: x in downarrow X;
    then reconsider x as Element of L1;
    reconsider x9 = x as Element of L2 by A1;
    consider y being Element of L1 such that
A4: y >= x and
A5: y in X by A3,Def15;
    reconsider y9 = y as Element of L2 by A1;
    y9 >= x9 by A1,A4,YELLOW_0:1;
    hence thesis by A2,A5,Def15;
  end;
  thus downarrow Y c= downarrow X
  proof
    let x be object;
    assume
A6: x in downarrow Y;
    then reconsider x as Element of L2;
    reconsider x9 = x as Element of L1 by A1;
    consider y being Element of L2 such that
A7: y >= x and
A8: y in Y by A6,Def15;
    reconsider y9 = y as Element of L1 by A1;
    y9 >= x9 by A1,A7,YELLOW_0:1;
    hence thesis by A2,A8,Def15;
  end;
  thus uparrow X c= uparrow Y
  proof
    let x be object;
    assume
A9: x in uparrow X;
    then reconsider x as Element of L1;
    reconsider x9 = x as Element of L2 by A1;
    consider y being Element of L1 such that
A10: y <= x and
A11: y in X by A9,Def16;
    reconsider y9 = y as Element of L2 by A1;
    y9 <= x9 by A1,A10,YELLOW_0:1;
    hence thesis by A2,A11,Def16;
  end;
  thus uparrow Y c= uparrow X
  proof
    let x be object;
    assume
A12: x in uparrow Y;
    then reconsider x as Element of L2;
    reconsider x9 = x as Element of L1 by A1;
    consider y being Element of L2 such that
A13: y <= x and
A14: y in Y by A12,Def16;
    reconsider y9 = y as Element of L1 by A1;
    y9 <= x9 by A1,A13,YELLOW_0:1;
    hence thesis by A2,A14,Def16;
  end;
end;
