
theorem Th4:
  for R being non empty RelStr for F being Subset of R holds
  uparrow F = union {uparrow x where x is Element of R: x in F} &
  downarrow F = union {downarrow x where x is Element of R: x in F}
proof
  let R be non empty RelStr, F be Subset of R;
A1: uparrow F = {x where x is Element of R:
  ex y being Element of R st x >= y & y in F} by WAYBEL_0:15;
A2: downarrow F = {x where x is Element of R:
  ex y being Element of R st x <= y & y in F} by WAYBEL_0:14;
  hereby
    let a be object;
    assume a in uparrow F;
    then consider x being Element of R such that
A3: a = x and
A4: ex y being Element of R st x >= y & y in F by A1;
    consider y being Element of R such that
A5: x >= y and
A6: y in F by A4;
A7: uparrow y in {uparrow z where z is Element of R: z in F} by A6;
    x in uparrow y by A5,WAYBEL_0:18;
    hence a in union {uparrow z where z is Element of R: z in F}
    by A3,A7,TARSKI:def 4;
  end;
  hereby
    let a be object;
    assume a in union {uparrow x where x is Element of R: x in F};
    then consider X being set such that
A8: a in X and
A9: X in {uparrow x where x is Element of R: x in F} by TARSKI:def 4;
    consider x being Element of R such that
A10: X = uparrow x and
A11: x in F by A9;
    reconsider y = a as Element of R by A8,A10;
    y >= x by A8,A10,WAYBEL_0:18;
    hence a in uparrow F by A1,A11;
  end;
  hereby
    let a be object;
    assume a in downarrow F;
    then consider x being Element of R such that
A12: a = x and
A13: ex y being Element of R st x <= y & y in F by A2;
    consider y being Element of R such that
A14: x <= y and
A15: y in F by A13;
A16: downarrow y in {downarrow z where z is Element of R: z in F} by A15;
    x in downarrow y by A14,WAYBEL_0:17;
    hence a in union {downarrow z where z is Element of R: z in F}
    by A12,A16,TARSKI:def 4;
  end;
  hereby
    let a be object;
    assume a in union {downarrow x where x is Element of R: x in F};
    then consider X being set such that
A17: a in X and
A18: X in {downarrow x where x is Element of R: x in F} by TARSKI:def 4;
    consider x being Element of R such that
A19: X = downarrow x and
A20: x in F by A18;
    reconsider y = a as Element of R by A17,A19;
    y <= x by A17,A19,WAYBEL_0:17;
    hence a in downarrow F by A2,A20;
  end;
end;
