Article FUZZY_2, MML version 4.99.1005

:: FUZZY_2:th 1
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Membership_Func of b1 holds
   0 <= b3 . b2 & b3 . b2 <= 1;

:: FUZZY_2:th 2
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than b3
   holds max(b2,min(b4,b3)) = min(max(b2,b4),b3);

:: FUZZY_2:funcnot 1 => FUZZY_2:func 1
definition
  let a1 be non empty set;
  let a2, a3 be Membership_Func of a1;
  func A2 \ A3 -> Membership_Func of a1 equals
    min(a2,1_minus a3);
end;

:: FUZZY_2:def 1
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
b2 \ b3 = min(b2,1_minus b3);

:: FUZZY_2:th 3
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
1_minus (b2 \ b3) = max(1_minus b2,b3);

:: FUZZY_2:th 4
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than b3
   holds b2 \ b4 is_less_than b3 \ b4;

:: FUZZY_2:th 5
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 is_less_than b3
   holds b4 \ b3 is_less_than b4 \ b2;

:: FUZZY_2:th 6
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Membership_Func of b1
      st b2 is_less_than b3 & b4 is_less_than b5
   holds b2 \ b5 is_less_than b3 \ b4;

:: FUZZY_2:th 9
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   b2 \ EMF b1 = b2;

:: FUZZY_2:th 11
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
b2 \ b3 is_less_than b2 \ min(b2,b3);

:: FUZZY_2:th 12
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
max(min(b2,b3),b2 \ b3) is_less_than b2;

:: FUZZY_2:th 13
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
max(b2,b3 \ b2) is_less_than max(b2,b3);

:: FUZZY_2:th 14
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
b2 \ (b3 \ b4) = max(b2 \ b3,min(b2,b4));

:: FUZZY_2:th 15
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
min(b2,b3) is_less_than b2 \ (b2 \ b3);

:: FUZZY_2:th 16
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
b2 \ b3 is_less_than (max(b2,b3)) \ b3;

:: FUZZY_2:th 17
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
b2 \ max(b3,b4) = min(b2 \ b3,b2 \ b4);

:: FUZZY_2:th 18
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
b2 \ min(b3,b4) = max(b2 \ b3,b2 \ b4);

:: FUZZY_2:th 19
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
(b2 \ b3) \ b4 = b2 \ max(b3,b4);

:: FUZZY_2:th 20
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
b2 \+\ b3 is_less_than (max(b2,b3)) \ min(b2,b3);

:: FUZZY_2:th 22
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1
      st b2 \ b3 is_less_than b4 & b3 \ b2 is_less_than b4
   holds b2 \+\ b3 is_less_than b4;

:: FUZZY_2:th 24
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
min(b2,b3 \ b4) is_less_than (min(b2,b3)) \ min(b2,b4);

:: FUZZY_2:th 25
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
b2 \+\ b3 is_less_than (max(b2,b3)) \ min(b2,b3);

:: FUZZY_2:th 26
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
max(min(b2,b3),1_minus max(b2,b3)) is_less_than 1_minus (b2 \+\ b3);

:: FUZZY_2:th 27
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
(b2 \+\ b3) \ b4 = max(b2 \ max(b3,b4),b3 \ max(b2,b4));

:: FUZZY_2:th 28
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
max(b2 \ max(b3,b4),min(min(b2,b3),b4)) is_less_than b2 \ (b3 \+\ b4);

:: FUZZY_2:th 29
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1
      st b2 is_less_than b3
   holds max(b2,b3 \ b2) is_less_than b3;

:: FUZZY_2:th 30
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
max(b2 \+\ b3,min(b2,b3)) is_less_than max(b2,b3);

:: FUZZY_2:th 31
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1
      st b2 \ b3 = EMF b1
   holds b2 is_less_than b3;

:: FUZZY_2:th 32
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1
      st min(b2,b3) = EMF b1
   holds b2 \ b3 = b2;

:: FUZZY_2:funcnot 2 => FUZZY_2:func 2
definition
  let a1 be non empty set;
  let a2, a3 be Membership_Func of a1;
  func A2 * A3 -> Membership_Func of a1 means
    for b1 being Element of a1 holds
       it . b1 = (a2 . b1) * (a3 . b1);
  commutativity;
::  for a1 being non empty set
::  for a2, a3 being Membership_Func of a1 holds
::  a2 * a3 = a3 * a2;
end;

:: FUZZY_2:def 2
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
   b4 = b2 * b3
iff
   for b5 being Element of b1 holds
      b4 . b5 = (b2 . b5) * (b3 . b5);

:: FUZZY_2:funcnot 3 => FUZZY_2:func 3
definition
  let a1 be non empty set;
  let a2, a3 be Membership_Func of a1;
  func A2 ++ A3 -> Membership_Func of a1 means
    for b1 being Element of a1 holds
       it . b1 = ((a2 . b1) + (a3 . b1)) - ((a2 . b1) * (a3 . b1));
  commutativity;
::  for a1 being non empty set
::  for a2, a3 being Membership_Func of a1 holds
::  a2 ++ a3 = a3 ++ a2;
end;

:: FUZZY_2:def 3
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
   b4 = b2 ++ b3
iff
   for b5 being Element of b1 holds
      b4 . b5 = ((b2 . b5) + (b3 . b5)) - ((b2 . b5) * (b3 . b5));

:: FUZZY_2:th 33
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   b2 * b2 is_less_than b2 & b2 is_less_than b2 ++ b2;

:: FUZZY_2:th 34
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
(b2 * b3) * b4 = b2 * (b3 * b4);

:: FUZZY_2:th 35
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
(b2 ++ b3) ++ b4 = b2 ++ (b3 ++ b4);

:: FUZZY_2:th 36
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
b2 * (b2 ++ b3) is_less_than b2 & b2 is_less_than b2 ++ (b2 * b3);

:: FUZZY_2:th 37
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
b2 * (b3 ++ b4) is_less_than (b2 * b3) ++ (b2 * b4);

:: FUZZY_2:th 38
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
(b2 ++ b3) * (b2 ++ b4) is_less_than b2 ++ (b3 * b4);

:: FUZZY_2:th 39
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
1_minus (b2 * b3) = (1_minus b2) ++ 1_minus b3;

:: FUZZY_2:th 40
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
1_minus (b2 ++ b3) = (1_minus b2) * 1_minus b3;

:: FUZZY_2:th 41
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
b2 ++ b3 = 1_minus ((1_minus b2) * 1_minus b3);

:: FUZZY_2:th 42
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   b2 * EMF b1 = EMF b1 & b2 * UMF b1 = b2;

:: FUZZY_2:th 43
theorem
for b1 being non empty set
for b2 being Membership_Func of b1 holds
   b2 ++ EMF b1 = b2 & b2 ++ UMF b1 = UMF b1;

:: FUZZY_2:th 44
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
b2 * b3 is_less_than min(b2,b3);

:: FUZZY_2:th 45
theorem
for b1 being non empty set
for b2, b3 being Membership_Func of b1 holds
max(b2,b3) is_less_than b2 ++ b3;

:: FUZZY_2:th 46
theorem
for b1, b2, b3 being Element of REAL
      st 0 <= b3
   holds b3 * max(b1,b2) = max(b3 * b1,b3 * b2) &
    b3 * min(b1,b2) = min(b3 * b1,b3 * b2);

:: FUZZY_2:th 47
theorem
for b1, b2, b3 being Element of REAL holds
b3 + max(b1,b2) = max(b3 + b1,b3 + b2) &
 b3 + min(b1,b2) = min(b3 + b1,b3 + b2);

:: FUZZY_2:th 48
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
b2 * max(b3,b4) = max(b2 * b3,b2 * b4);

:: FUZZY_2:th 49
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
b2 * min(b3,b4) = min(b2 * b3,b2 * b4);

:: FUZZY_2:th 50
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
b2 ++ max(b3,b4) = max(b2 ++ b3,b2 ++ b4);

:: FUZZY_2:th 51
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
b2 ++ min(b3,b4) = min(b2 ++ b3,b2 ++ b4);

:: FUZZY_2:th 52
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
(max(b2,b3)) * max(b2,b4) is_less_than max(b2,b3 * b4);

:: FUZZY_2:th 53
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
(min(b2,b3)) * min(b2,b4) is_less_than min(b2,b3 * b4);

:: FUZZY_2:th 54
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Membership_Func of b1 holds
(b3 ++ b4) . b2 = 1 - ((1 - (b3 . b2)) * (1 - (b4 . b2)));

:: FUZZY_2:th 55
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
max(b2,b3 ++ b4) is_less_than (max(b2,b3)) ++ max(b2,b4);

:: FUZZY_2:th 56
theorem
for b1 being non empty set
for b2, b3, b4 being Membership_Func of b1 holds
min(b2,b3 ++ b4) is_less_than (min(b2,b3)) ++ min(b2,b4);

:: FUZZY_2:condreg 1
registration
  let a1 be non empty set;
  cluster -> quasi_total (Membership_Func of a1);
end;

:: FUZZY_2:modenot 1
definition
  let a1, a2 be non empty set;
  mode RMembership_Func of a1,a2 is Membership_Func of [:a1,a2:];
end;

:: FUZZY_2:funcnot 4 => FUZZY_2:func 4
definition
  let a1, a2 be non empty set;
  func Zmf(A1,A2) -> Membership_Func of [:a1,a2:] equals
    chi({},[:a1,a2:]);
end;

:: FUZZY_2:def 4
theorem
for b1, b2 being non empty set holds
Zmf(b1,b2) = chi({},[:b1,b2:]);

:: FUZZY_2:funcnot 5 => FUZZY_2:func 5
definition
  let a1, a2 be non empty set;
  func Umf(A1,A2) -> Membership_Func of [:a1,a2:] equals
    chi([:a1,a2:],[:a1,a2:]);
end;

:: FUZZY_2:def 5
theorem
for b1, b2 being non empty set holds
Umf(b1,b2) = chi([:b1,b2:],[:b1,b2:]);

:: FUZZY_2:th 57
theorem
for b1, b2 being non empty set
for b3 being Element of [:b1,b2:]
for b4 being Membership_Func of [:b1,b2:] holds
   (Zmf(b1,b2)) . b3 <= b4 . b3 & b4 . b3 <= (Umf(b1,b2)) . b3;

:: FUZZY_2:th 58
theorem
for b1, b2 being non empty set
for b3 being Membership_Func of [:b1,b2:] holds
   max(b3,Umf(b1,b2)) = Umf(b1,b2) & min(b3,Umf(b1,b2)) = b3 & max(b3,Zmf(b1,b2)) = b3 & min(b3,Zmf(b1,b2)) = Zmf(b1,b2);

:: FUZZY_2:th 59
theorem
for b1, b2 being non empty set holds
1_minus Zmf(b1,b2) = Umf(b1,b2) & 1_minus Umf(b1,b2) = Zmf(b1,b2);

:: FUZZY_2:th 60
theorem
for b1, b2 being non empty set
for b3, b4 being Membership_Func of [:b1,b2:]
      st b3 \ b4 = Zmf(b1,b2)
   holds b3 is_less_than b4;

:: FUZZY_2:th 61
theorem
for b1, b2 being non empty set
for b3, b4 being Membership_Func of [:b1,b2:]
      st min(b3,b4) = Zmf(b1,b2)
   holds b3 \ b4 = b3;