Article YELLOW_5, MML version 4.99.1005

:: YELLOW_5:th 1
theorem
for b1 being reflexive antisymmetric with_suprema RelStr
for b2 being Element of the carrier of b1 holds
   b2 "\/" b2 = b2;

:: YELLOW_5:th 2
theorem
for b1 being reflexive antisymmetric with_infima RelStr
for b2 being Element of the carrier of b1 holds
   b2 "/\" b2 = b2;

:: YELLOW_5:th 3
theorem
for b1 being transitive antisymmetric with_suprema RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 "\/" b3 <= b4
   holds b2 <= b4;

:: YELLOW_5:th 4
theorem
for b1 being transitive antisymmetric with_infima RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b4 <= b2 "/\" b3
   holds b4 <= b2;

:: YELLOW_5:th 5
theorem
for b1 being transitive antisymmetric with_suprema with_infima RelStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 "/\" b3 <= b2 "\/" b4;

:: YELLOW_5:th 6
theorem
for b1 being transitive antisymmetric with_infima RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b3
   holds b2 "/\" b4 <= b3 "/\" b4;

:: YELLOW_5:th 7
theorem
for b1 being transitive antisymmetric with_suprema RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b3
   holds b2 "\/" b4 <= b3 "\/" b4;

:: YELLOW_5:th 8
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2, b3 being Element of the carrier of b1
      st b2 <= b3
   holds b2 "\/" b3 = b3;

:: YELLOW_5:th 9
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b4 & b3 <= b4
   holds b2 "\/" b3 <= b4;

:: YELLOW_5:th 10
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2, b3 being Element of the carrier of b1
      st b3 <= b2
   holds b2 "/\" b3 = b3;

:: YELLOW_5:th 11
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
   b3 is_a_complement_of b2
iff
   b3 = 'not' b2;

:: YELLOW_5:funcnot 1 => YELLOW_5:func 1
definition
  let a1 be non empty RelStr;
  let a2, a3 be Element of the carrier of a1;
  func A2 \ A3 -> Element of the carrier of a1 equals
    a2 "/\" 'not' a3;
end;

:: YELLOW_5:def 1
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 \ b3 = b2 "/\" 'not' b3;

:: YELLOW_5:funcnot 2 => YELLOW_5:func 2
definition
  let a1 be non empty RelStr;
  let a2, a3 be Element of the carrier of a1;
  func A2 \+\ A3 -> Element of the carrier of a1 equals
    (a2 \ a3) "\/" (a3 \ a2);
end;

:: YELLOW_5:def 2
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 \+\ b3 = (b2 \ b3) "\/" (b3 \ b2);

:: YELLOW_5:funcnot 3 => YELLOW_5:func 3
definition
  let a1 be antisymmetric with_suprema with_infima RelStr;
  let a2, a3 be Element of the carrier of a1;
  redefine func a2 \+\ a3 -> Element of the carrier of a1;
  commutativity;
::  for a1 being antisymmetric with_suprema with_infima RelStr
::  for a2, a3 being Element of the carrier of a1 holds
::  a2 \+\ a3 = a3 \+\ a2;
end;

:: YELLOW_5:prednot 1 => YELLOW_5:pred 1
definition
  let a1 be non empty RelStr;
  let a2, a3 be Element of the carrier of a1;
  pred A2 meets A3 means
    a2 "/\" a3 <> Bottom a1;
end;

:: YELLOW_5:dfs 3
definiens
  let a1 be non empty RelStr;
  let a2, a3 be Element of the carrier of a1;
To prove
     a2 meets a3
it is sufficient to prove
  thus a2 "/\" a3 <> Bottom a1;

:: YELLOW_5:def 3
theorem
for b1 being non empty RelStr
for b2, b3 being Element of the carrier of b1 holds
   b2 meets b3
iff
   b2 "/\" b3 <> Bottom b1;

:: YELLOW_5:prednot 2 => not YELLOW_5:pred 1
notation
  let a1 be non empty RelStr;
  let a2, a3 be Element of the carrier of a1;
  antonym a2 misses a3 for a2 meets a3;
end;

:: YELLOW_5:prednot 3 => not YELLOW_5:pred 1
notation
  let a1 be antisymmetric with_infima RelStr;
  let a2, a3 be Element of the carrier of a1;
  antonym a2 misses a3 for a2 meets a3;
end;

:: YELLOW_5:prednot 4 => YELLOW_5:pred 2
definition
  let a1 be antisymmetric with_infima RelStr;
  let a2, a3 be Element of the carrier of a1;
  redefine pred a2 meets a3;
  symmetry;
::  for a1 being antisymmetric with_infima RelStr
::  for a2, a3 being Element of the carrier of a1
::        st a2 meets a3
::     holds a3 meets a2;
end;

:: YELLOW_5:th 12
theorem
for b1 being transitive antisymmetric with_suprema with_infima RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b4
   holds b2 \ b3 <= b4;

:: YELLOW_5:th 13
theorem
for b1 being transitive antisymmetric with_suprema with_infima RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b3
   holds b2 \ b4 <= b3 \ b4;

:: YELLOW_5:th 16
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 \ b3 <= b4 & b3 \ b2 <= b4
   holds b2 \+\ b3 <= b4;

:: YELLOW_5:th 17
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1 holds
      b2 meets b2
   iff
      b2 <> Bottom b1;

:: YELLOW_5:th 19
theorem
for b1 being transitive antisymmetric with_infima RelStr
   st b1 is distributive
for b2, b3, b4 being Element of the carrier of b1
      st (b2 "/\" b3) "\/" (b2 "/\" b4) = b2
   holds b2 <= b3 "\/" b4;

:: YELLOW_5:th 20
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
   st b1 is distributive
for b2, b3, b4 being Element of the carrier of b1 holds
b2 "\/" (b3 "/\" b4) = (b2 "\/" b3) "/\" (b2 "\/" b4);

:: YELLOW_5:th 21
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
   st b1 is distributive
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 "\/" b3) \ b4 = (b2 \ b4) "\/" (b3 \ b4);

:: YELLOW_5:th 22
theorem
for b1 being non empty antisymmetric lower-bounded RelStr
for b2 being Element of the carrier of b1
      st b2 <= Bottom b1
   holds b2 = Bottom b1;

:: YELLOW_5:th 23
theorem
for b1 being reflexive transitive antisymmetric with_infima lower-bounded RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b3 & b2 <= b4 & b3 "/\" b4 = Bottom b1
   holds b2 = Bottom b1;

:: YELLOW_5:th 24
theorem
for b1 being antisymmetric with_suprema lower-bounded RelStr
for b2, b3 being Element of the carrier of b1
      st b2 "\/" b3 = Bottom b1
   holds b2 = Bottom b1 & b3 = Bottom b1;

:: YELLOW_5:th 25
theorem
for b1 being transitive antisymmetric with_infima lower-bounded RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b3 & b3 "/\" b4 = Bottom b1
   holds b2 "/\" b4 = Bottom b1;

:: YELLOW_5:th 26
theorem
for b1 being reflexive transitive antisymmetric with_infima lower-bounded RelStr
for b2 being Element of the carrier of b1 holds
   (Bottom b1) \ b2 = Bottom b1;

:: YELLOW_5:th 27
theorem
for b1 being transitive antisymmetric with_infima lower-bounded RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 meets b3 & b3 <= b4
   holds b2 meets b4;

:: YELLOW_5:th 28
theorem
for b1 being antisymmetric with_infima lower-bounded RelStr
for b2 being Element of the carrier of b1 holds
   b2 "/\" Bottom b1 = Bottom b1;

:: YELLOW_5:th 29
theorem
for b1 being transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 meets b3 "/\" b4
   holds b2 meets b3;

:: YELLOW_5:th 30
theorem
for b1 being transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 meets b3 \ b4
   holds b2 meets b3;

:: YELLOW_5:th 31
theorem
for b1 being transitive antisymmetric with_infima lower-bounded RelStr
for b2 being Element of the carrier of b1 holds
   b2 misses Bottom b1;

:: YELLOW_5:th 32
theorem
for b1 being transitive antisymmetric with_infima lower-bounded RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 misses b4 & b3 <= b4
   holds b2 misses b3;

:: YELLOW_5:th 33
theorem
for b1 being transitive antisymmetric with_infima lower-bounded RelStr
for b2, b3, b4 being Element of the carrier of b1
      st (b2 meets b3 implies b2 misses b4)
   holds b2 misses b3 "/\" b4;

:: YELLOW_5:th 34
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima lower-bounded RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b3 & b2 <= b4 & b3 misses b4
   holds b2 = Bottom b1;

:: YELLOW_5:th 35
theorem
for b1 being transitive antisymmetric with_infima lower-bounded RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 misses b3
   holds b2 "/\" b4 misses b3 "/\" b4;

:: YELLOW_5:th 36
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4 being Element of the carrier of b1 holds
((b2 "/\" b3) "\/" (b3 "/\" b4)) "\/" (b4 "/\" b2) = ((b2 "\/" b3) "/\" (b3 "\/" b4)) "/\" (b4 "\/" b2);

:: YELLOW_5:th 37
theorem
for b1 being non empty Boolean RelStr
for b2 being Element of the carrier of b1 holds
   b2 "/\" 'not' b2 = Bottom b1 & b2 "\/" 'not' b2 = Top b1;

:: YELLOW_5:th 38
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 \ b3 <= b4
   holds b2 <= b3 "\/" b4;

:: YELLOW_5:th 39
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
'not' (b2 "\/" b3) = ('not' b2) "/\" 'not' b3 &
 'not' (b2 "/\" b3) = ('not' b2) "\/" 'not' b3;

:: YELLOW_5:th 40
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1
      st b2 <= b3
   holds 'not' b3 <= 'not' b2;

:: YELLOW_5:th 41
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b3
   holds b4 \ b3 <= b4 \ b2;

:: YELLOW_5:th 42
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4, b5 being Element of the carrier of b1
      st b2 <= b3 & b4 <= b5
   holds b2 \ b5 <= b3 \ b4;

:: YELLOW_5:th 43
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b3 "\/" b4
   holds b2 \ b3 <= b4 & b2 \ b4 <= b3;

:: YELLOW_5:th 44
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
'not' b2 <= 'not' (b2 "/\" b3) & 'not' b3 <= 'not' (b2 "/\" b3);

:: YELLOW_5:th 45
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
'not' (b2 "\/" b3) <= 'not' b2 & 'not' (b2 "\/" b3) <= 'not' b3;

:: YELLOW_5:th 46
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1
      st b2 <= b3 \ b2
   holds b2 = Bottom b1;

:: YELLOW_5:th 47
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1
      st b2 <= b3
   holds b3 = b2 "\/" (b3 \ b2);

:: YELLOW_5:th 48
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
   b2 \ b3 = Bottom b1
iff
   b2 <= b3;

:: YELLOW_5:th 49
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4 being Element of the carrier of b1
      st b2 <= b3 "\/" b4 & b2 "/\" b4 = Bottom b1
   holds b2 <= b3;

:: YELLOW_5:th 50
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 "\/" b3 = (b2 \ b3) "\/" b3;

:: YELLOW_5:th 51
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b2 "\/" b3) = Bottom b1;

:: YELLOW_5:th 52
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b2 "/\" b3) = b2 \ b3;

:: YELLOW_5:th 53
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
(b2 \ b3) "/\" b3 = Bottom b1;

:: YELLOW_5:th 54
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 "\/" (b3 \ b2) = b2 "\/" b3;

:: YELLOW_5:th 55
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
(b2 "/\" b3) "\/" (b2 \ b3) = b2;

:: YELLOW_5:th 56
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 \ (b3 \ b4) = (b2 \ b3) "\/" (b2 "/\" b4);

:: YELLOW_5:th 57
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 \ (b2 \ b3) = b2 "/\" b3;

:: YELLOW_5:th 58
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
(b2 "\/" b3) \ b3 = b2 \ b3;

:: YELLOW_5:th 59
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
   b2 "/\" b3 = Bottom b1
iff
   b2 \ b3 = b2;

:: YELLOW_5:th 60
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 \ (b3 "\/" b4) = (b2 \ b3) "/\" (b2 \ b4);

:: YELLOW_5:th 61
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 \ (b3 "/\" b4) = (b2 \ b3) "\/" (b2 \ b4);

:: YELLOW_5:th 62
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 "/\" (b3 \ b4) = (b2 "/\" b3) \ (b2 "/\" b4);

:: YELLOW_5:th 63
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
(b2 "\/" b3) \ (b2 "/\" b3) = (b2 \ b3) "\/" (b3 \ b2);

:: YELLOW_5:th 64
theorem
for b1 being non empty Boolean RelStr
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 \ b3) \ b4 = b2 \ (b3 "\/" b4);

:: YELLOW_5:th 65
theorem
for b1 being non empty Boolean RelStr holds
   'not' Bottom b1 = Top b1;

:: YELLOW_5:th 66
theorem
for b1 being non empty Boolean RelStr holds
   'not' Top b1 = Bottom b1;

:: YELLOW_5:th 67
theorem
for b1 being non empty Boolean RelStr
for b2 being Element of the carrier of b1 holds
   b2 \ b2 = Bottom b1;

:: YELLOW_5:th 68
theorem
for b1 being non empty Boolean RelStr
for b2 being Element of the carrier of b1 holds
   b2 \ Bottom b1 = b2;

:: YELLOW_5:th 69
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
'not' (b2 \ b3) = ('not' b2) "\/" b3;

:: YELLOW_5:th 70
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 "/\" b3 misses b2 \ b3;

:: YELLOW_5:th 71
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 \ b3 misses b3;

:: YELLOW_5:th 72
theorem
for b1 being non empty Boolean RelStr
for b2, b3 being Element of the carrier of b1
      st b2 misses b3
   holds (b2 "\/" b3) \ b3 = b2;