Article REALSET1, MML version 4.99.1005

:: REALSET1:th 1
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1
      st b2 in [:b1,b1:]
   holds b3 . b2 in b1;

:: REALSET1:th 2
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1 holds
   ex b3 being Element of bool b1 st
      for b4 being set
            st b4 in [:b3,b3:]
         holds b2 . b4 in b3;

:: REALSET1:prednot 1 => REALSET1:pred 1
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a3 be Element of bool a1;
  pred A2 is_in A3 means
    for b1 being set
          st b1 in [:a3,a3:]
       holds a2 . b1 in a3;
end;

:: REALSET1:dfs 1
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a3 be Element of bool a1;
To prove
     a2 is_in a3
it is sufficient to prove
  thus for b1 being set
          st b1 in [:a3,a3:]
       holds a2 . b1 in a3;

:: REALSET1:def 1
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
for b3 being Element of bool b1 holds
      b2 is_in b3
   iff
      for b4 being set
            st b4 in [:b3,b3:]
         holds b2 . b4 in b3;

:: REALSET1:modenot 1 => REALSET1:mode 1
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
  mode Preserv of A2 -> Element of bool a1 means
    for b1 being set
          st b1 in [:it,it:]
       holds a2 . b1 in it;
end;

:: REALSET1:dfs 2
definiens
  let a1 be set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a3 be Element of bool a1;
To prove
     a3 is Preserv of a2
it is sufficient to prove
  thus for b1 being set
          st b1 in [:a3,a3:]
       holds a2 . b1 in a3;

:: REALSET1:def 2
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
for b3 being Element of bool b1 holds
      b3 is Preserv of b2
   iff
      for b4 being set
            st b4 in [:b3,b3:]
         holds b2 . b4 in b3;

:: REALSET1:funcnot 1 => REALSET1:func 1
definition
  let a1 be Relation-like set;
  let a2 be set;
  func A1 || A2 -> set equals
    a1 | [:a2,a2:];
end;

:: REALSET1:def 3
theorem
for b1 being Relation-like set
for b2 being set holds
   b1 || b2 = b1 | [:b2,b2:];

:: REALSET1:funcreg 1
registration
  let a1 be Relation-like set;
  let a2 be set;
  cluster a1 || a2 -> Relation-like;
end;

:: REALSET1:funcreg 2
registration
  let a1 be Relation-like Function-like set;
  let a2 be set;
  cluster a1 || a2 -> Function-like;
end;

:: REALSET1:th 3
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
for b3 being Preserv of b2 holds
   b2 || b3 is Function-like quasi_total Relation of [:b3,b3:],b3;

:: REALSET1:funcnot 2 => REALSET1:func 2
definition
  let a1 be set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a3 be Preserv of a2;
  redefine func a2 || a3 -> Function-like quasi_total Relation of [:a3,a3:],a3;
end;

:: REALSET1:attrnot 1 => REALSET1:attr 1
definition
  let a1 be set;
  attr a1 is trivial means
    (a1 is not empty) implies ex b1 being set st
       a1 = {b1};
end;

:: REALSET1:dfs 4
definiens
  let a1 be set;
To prove
     a1 is trivial
it is sufficient to prove
  thus (a1 is not empty) implies ex b1 being set st
       a1 = {b1};

:: REALSET1:def 4
theorem
for b1 being set holds
      b1 is trivial
   iff
      (b1 is not empty implies ex b2 being set st
         b1 = {b2});

:: REALSET1:exreg 1
registration
  cluster non empty trivial set;
end;

:: REALSET1:exreg 2
registration
  cluster non empty non trivial set;
end;

:: REALSET1:condreg 1
registration
  cluster non trivial -> non empty (set);
end;

:: REALSET1:funcreg 3
registration
  let a1 be set;
  cluster {a1} -> trivial;
end;

:: REALSET1:th 4
theorem
for b1 being non empty set holds
      b1 is not trivial
   iff
      for b2 being set holds
         b1 \ {b2} is non empty set;

:: REALSET1:th 5
theorem
ex b1 being non empty set st
   for b2 being Element of b1 holds
      b1 \ {b2} is non empty set;

:: REALSET1:th 6
theorem
for b1 being non empty set
      st for b2 being Element of b1 holds
           b1 \ {b2} is non empty set
   holds b1 is not trivial;

:: REALSET1:th 7
theorem
for b1 being non empty set
      st for b2 being Element of b1 holds
           b1 \ {b2} is non empty set
   holds b1 is non trivial set;

:: REALSET1:prednot 2 => REALSET1:pred 2
definition
  let a1 be non trivial set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a3 be Element of a1;
  pred A2 is_Bin_Op_Preserv A3 means
    a1 \ {a3} is Preserv of a2 &
     (a2 || a1) \ {a3} is Function-like quasi_total Relation of [:a1 \ {a3},a1 \ {a3}:],a1 \ {a3};
end;

:: REALSET1:dfs 5
definiens
  let a1 be non trivial set;
  let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
  let a3 be Element of a1;
To prove
     a2 is_Bin_Op_Preserv a3
it is sufficient to prove
  thus a1 \ {a3} is Preserv of a2 &
     (a2 || a1) \ {a3} is Function-like quasi_total Relation of [:a1 \ {a3},a1 \ {a3}:],a1 \ {a3};

:: REALSET1:def 5
theorem
for b1 being non trivial set
for b2 being Function-like quasi_total Relation of [:b1,b1:],b1
for b3 being Element of b1 holds
      b2 is_Bin_Op_Preserv b3
   iff
      b1 \ {b3} is Preserv of b2 &
       (b2 || b1) \ {b3} is Function-like quasi_total Relation of [:b1 \ {b3},b1 \ {b3}:],b1 \ {b3};

:: REALSET1:th 8
theorem
for b1 being set
for b2 being Element of bool b1 holds
   ex b3 being Function-like quasi_total Relation of [:b1,b1:],b1 st
      for b4 being set
            st b4 in [:b2,b2:]
         holds b3 . b4 in b2;

:: REALSET1:modenot 2 => REALSET1:mode 2
definition
  let a1 be set;
  let a2 be Element of bool a1;
  mode Presv of A1,A2 -> Function-like quasi_total Relation of [:a1,a1:],a1 means
    for b1 being set
          st b1 in [:a2,a2:]
       holds it . b1 in a2;
end;

:: REALSET1:dfs 6
definiens
  let a1 be set;
  let a2 be Element of bool a1;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
To prove
     a3 is Presv of a1,a2
it is sufficient to prove
  thus for b1 being set
          st b1 in [:a2,a2:]
       holds a3 . b1 in a2;

:: REALSET1:def 6
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1 holds
      b3 is Presv of b1,b2
   iff
      for b4 being set
            st b4 in [:b2,b2:]
         holds b3 . b4 in b2;

:: REALSET1:th 9
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Presv of b1,b2 holds
   b3 || b2 is Function-like quasi_total Relation of [:b2,b2:],b2;

:: REALSET1:funcnot 3 => REALSET1:func 3
definition
  let a1 be set;
  let a2 be Element of bool a1;
  let a3 be Presv of a1,a2;
  func A3 ||| A2 -> Function-like quasi_total Relation of [:a2,a2:],a2 equals
    a3 || a2;
end;

:: REALSET1:def 7
theorem
for b1 being set
for b2 being Element of bool b1
for b3 being Presv of b1,b2 holds
   b3 ||| b2 = b3 || b2;

:: REALSET1:modenot 3 => REALSET1:mode 3
definition
  let a1 be set;
  let a2 be Element of a1;
  mode DnT of A2,A1 -> Function-like quasi_total Relation of [:a1,a1:],a1 means
    for b1 being set
          st b1 in [:a1 \ {a2},a1 \ {a2}:]
       holds it . b1 in a1 \ {a2};
end;

:: REALSET1:dfs 8
definiens
  let a1 be set;
  let a2 be Element of a1;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],a1;
To prove
     a3 is DnT of a2,a1
it is sufficient to prove
  thus for b1 being set
          st b1 in [:a1 \ {a2},a1 \ {a2}:]
       holds a3 . b1 in a1 \ {a2};

:: REALSET1:def 8
theorem
for b1 being set
for b2 being Element of b1
for b3 being Function-like quasi_total Relation of [:b1,b1:],b1 holds
      b3 is DnT of b2,b1
   iff
      for b4 being set
            st b4 in [:b1 \ {b2},b1 \ {b2}:]
         holds b3 . b4 in b1 \ {b2};

:: REALSET1:th 11
theorem
for b1 being non trivial set
for b2 being Element of b1
for b3 being DnT of b2,b1 holds
   b3 || (b1 \ {b2}) is Function-like quasi_total Relation of [:b1 \ {b2},b1 \ {b2}:],b1 \ {b2};

:: REALSET1:funcnot 4 => REALSET1:func 4
definition
  let a1 be non trivial set;
  let a2 be Element of a1;
  let a3 be DnT of a2,a1;
  func A3 !(A1,A2) -> Function-like quasi_total Relation of [:a1 \ {a2},a1 \ {a2}:],a1 \ {a2} equals
    a3 || (a1 \ {a2});
end;

:: REALSET1:def 9
theorem
for b1 being non trivial set
for b2 being Element of b1
for b3 being DnT of b2,b1 holds
   b3 !(b1,b2) = b3 || (b1 \ {b2});

:: REALSET1:exreg 3
registration
  let a1 be non empty set;
  cluster non empty trivial Element of bool a1;
end;

:: REALSET1:modenot 4
definition
  let a1 be non empty set;
  mode OnePoint of a1 is non empty trivial Element of bool a1;
end;

:: REALSET1:th 12
theorem
for b1 being non trivial set
for b2 being non empty trivial Element of bool b1 holds
   b1 \ b2 is non empty set;

:: REALSET1:funcreg 4
registration
  let a1 be non trivial set;
  let a2 be non empty trivial Element of bool a1;
  cluster a1 \ a2 -> non empty;
end;

:: REALSET1:th 13
theorem
for b1 being finite set holds
      b1 is trivial
   iff
      card b1 < 2;

:: REALSET1:th 14
theorem
for b1 being set holds
      b1 is not trivial
   iff
      ex b2, b3 being set st
         b2 in b1 & b3 in b1 & b2 <> b3;

:: REALSET1:th 15
theorem
for b1 being set
for b2 being Element of bool b1 holds
      b2 is not trivial
   iff
      ex b3, b4 being Element of b1 st
         b3 in b2 & b4 in b2 & b3 <> b4;

:: REALSET1:th 16
theorem
for b1 being non trivial set
for b2 being non empty trivial Element of bool b1 holds
   ex b3 being Element of b1 st
      b2 = {b3};

:: REALSET1:funcreg 5
registration
  let a1, a2 be set;
  cluster a1 .--> a2 -> trivial;
end;

:: REALSET1:condreg 2
registration
  cluster empty -> trivial (set);
end;

:: REALSET1:condreg 3
registration
  cluster trivial -> finite (set);
end;

:: REALSET1:condreg 4
registration
  cluster infinite -> non trivial (set);
end;