Article YELLOW14, MML version 4.99.1005

:: YELLOW14:th 1
theorem
bool 1 = {0,1};

:: YELLOW14:th 2
theorem
for b1 being set
for b2 being Element of bool b1 holds
   rng ((id b1) | b2) = b2;

:: YELLOW14:exreg 1
registration
  cluster empty strict RelStr;
end;

:: YELLOW14:th 4
theorem
for b1 being empty 1-sorted
for b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st rng b3 = [#] b2
   holds b2 is empty;

:: YELLOW14:th 5
theorem
for b1 being 1-sorted
for b2 being empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st dom b3 = [#] b1
   holds b1 is empty;

:: YELLOW14:th 6
theorem
for b1 being non empty 1-sorted
for b2 being 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st dom b3 = [#] b1
   holds b2 is not empty;

:: YELLOW14:th 7
theorem
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st rng b3 = [#] b2
   holds b1 is not empty;

:: YELLOW14:attrnot 1 => WAYBEL_0:attr 22
definition
  let a1, a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is directed-sups-preserving means
    for b1 being non empty directed Element of bool the carrier of a1 holds
       a3 preserves_sup_of b1;
end;

:: YELLOW14:dfs 1
definiens
  let a1 be non empty reflexive RelStr;
  let a2 be non empty RelStr;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is directed-sups-preserving
it is sufficient to prove
  thus for b1 being non empty directed Element of bool the carrier of a1 holds
       a3 preserves_sup_of b1;

:: YELLOW14:def 1
theorem
for b1 being non empty reflexive RelStr
for b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is directed-sups-preserving(b1, b2)
   iff
      for b4 being non empty directed Element of bool the carrier of b1 holds
         b3 preserves_sup_of b4;

:: YELLOW14:attrnot 2 => YELLOW14:attr 1
definition
  let a1 be 1-sorted;
  let a2 be NetStr over a1;
  attr a2 is Function-yielding means
    the mapping of a2 is Function-yielding;
end;

:: YELLOW14:dfs 2
definiens
  let a1 be 1-sorted;
  let a2 be NetStr over a1;
To prove
     a2 is Function-yielding
it is sufficient to prove
  thus the mapping of a2 is Function-yielding;

:: YELLOW14:def 2
theorem
for b1 being 1-sorted
for b2 being NetStr over b1 holds
      b2 is Function-yielding(b1)
   iff
      the mapping of b2 is Function-yielding;

:: YELLOW14:exreg 2
registration
  cluster non empty constituted-Functions 1-sorted;
end;

:: YELLOW14:exreg 3
registration
  cluster non empty strict constituted-Functions RelStr;
end;

:: YELLOW14:condreg 1
registration
  let a1 be constituted-Functions 1-sorted;
  cluster -> Function-yielding (NetStr over a1);
end;

:: YELLOW14:exreg 4
registration
  let a1 be constituted-Functions 1-sorted;
  cluster strict Function-yielding NetStr over a1;
end;

:: YELLOW14:exreg 5
registration
  let a1 be non empty constituted-Functions 1-sorted;
  cluster non empty strict Function-yielding NetStr over a1;
end;

:: YELLOW14:funcreg 1
registration
  let a1 be constituted-Functions 1-sorted;
  let a2 be Function-yielding NetStr over a1;
  cluster the mapping of a2 -> Function-like quasi_total Function-yielding;
end;

:: YELLOW14:exreg 6
registration
  let a1 be non empty constituted-Functions 1-sorted;
  cluster non empty transitive strict directed Function-yielding NetStr over a1;
end;

:: YELLOW14:funcreg 2
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty NetStr over a1;
  cluster proj2 the mapping of a2 -> non empty;
end;

:: YELLOW14:funcreg 3
registration
  let a1 be non empty 1-sorted;
  let a2 be non empty NetStr over a1;
  cluster proj2 netmap(a2,a1) -> non empty;
end;

:: YELLOW14:th 8
theorem
for b1, b2, b3 being non empty RelStr
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b3
for b5, b6 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b5 <= b6 & b4 is monotone(b2, b3)
   holds b4 * b5 <= b4 * b6;

:: YELLOW14:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty TopSpace-like TopRelStr
for b3, b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5, b6 being Element of the carrier of ContMaps(b1,b2)
      st b5 = b3 & b6 = b4
   holds    b5 <= b6
   iff
      b3 <= b4;

:: YELLOW14:funcnot 1 => YELLOW14:func 1
definition
  let a1 be non empty set;
  let a2 be non empty RelStr;
  let a3 be Element of the carrier of a2 |^ a1;
  let a4 be Element of a1;
  redefine func a3 . a4 -> Element of the carrier of a2;
end;

:: YELLOW14:th 10
theorem
for b1, b2 being RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is isomorphic(b1, b2)
   holds b3 is onto(the carrier of b1, the carrier of b2);

:: YELLOW14:condreg 2
registration
  let a1, a2 be RelStr;
  cluster Function-like quasi_total isomorphic -> onto (Relation of the carrier of a1,the carrier of a2);
end;

:: YELLOW14:th 11
theorem
for b1, b2 being non empty RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is isomorphic(b1, b2)
   holds b3 /" is isomorphic(b2, b1);

:: YELLOW14:th 12
theorem
for b1, b2 being non empty RelStr
      st b1,b2 are_isomorphic & b1 is with_infima
   holds b2 is with_infima;

:: YELLOW14:th 13
theorem
for b1, b2 being non empty RelStr
      st b1,b2 are_isomorphic & b1 is with_suprema
   holds b2 is with_suprema;

:: YELLOW14:th 14
theorem
for b1 being RelStr
      st b1 is empty
   holds b1 is bounded;

:: YELLOW14:condreg 3
registration
  cluster empty -> bounded (RelStr);
end;

:: YELLOW14:th 15
theorem
for b1, b2 being RelStr
      st b1,b2 are_isomorphic & b1 is lower-bounded
   holds b2 is lower-bounded;

:: YELLOW14:th 16
theorem
for b1, b2 being RelStr
      st b1,b2 are_isomorphic & b1 is upper-bounded
   holds b2 is upper-bounded;

:: YELLOW14:th 17
theorem
for b1, b2 being non empty RelStr
for b3 being Element of bool the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b4 is isomorphic(b1, b2) & ex_sup_of b3,b1
   holds ex_sup_of b4 .: b3,b2;

:: YELLOW14:th 18
theorem
for b1, b2 being non empty RelStr
for b3 being Element of bool the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b4 is isomorphic(b1, b2) & ex_inf_of b3,b1
   holds ex_inf_of b4 .: b3,b2;

:: YELLOW14:th 19
theorem
for b1, b2 being TopStruct
      st (not b1,b2 are_homeomorphic implies ex b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 st
           dom b3 = [#] b1 & rng b3 = [#] b2)
   holds    b1 is empty
   iff
      b2 is empty;

:: YELLOW14:th 20
theorem
for b1 being non empty TopSpace-like TopStruct holds
   b1,TopStruct(#the carrier of b1,the topology of b1#) are_homeomorphic;

:: YELLOW14:condreg 4
registration
  let a1 be non empty reflexive Scott TopRelStr;
  cluster open -> upper inaccessible_by_directed_joins (Element of bool the carrier of a1);
end;

:: YELLOW14:condreg 5
registration
  let a1 be non empty reflexive Scott TopRelStr;
  cluster upper inaccessible_by_directed_joins -> open (Element of bool the carrier of a1);
end;

:: YELLOW14:th 21
theorem
for b1 being TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
      st b4 = {b2} & Cl b4 c= Cl b5
   holds b2 in Cl b5;

:: YELLOW14:th 22
theorem
for b1 being TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
      st b4 = {b3} & b2 in Cl b4 & b5 is open(b1) & b2 in b5
   holds b3 in b5;

:: YELLOW14:th 23
theorem
for b1 being TopStruct
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1
      st b4 = {b2} &
         b5 = {b3} &
         (for b6 being Element of bool the carrier of b1
               st b6 is open(b1) & b2 in b6
            holds b3 in b6)
   holds Cl b4 c= Cl b5;

:: YELLOW14:th 24
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being irreducible Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#)
   holds b4 is irreducible(b2);

:: YELLOW14:th 25
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of b2
for b5 being Element of bool the carrier of b1
for b6 being Element of bool the carrier of b2
      st b3 = b4 &
         b5 = b6 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b3 is_dense_point_of b5
   holds b4 is_dense_point_of b6;

:: YELLOW14:th 26
theorem
for b1, b2 being TopStruct
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4 &
         TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b3 is compact(b1)
   holds b4 is compact(b2);

:: YELLOW14:th 27
theorem
for b1, b2 being non empty TopSpace-like TopStruct
      st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b1 is sober
   holds b2 is sober;

:: YELLOW14:th 28
theorem
for b1, b2 being non empty TopSpace-like TopStruct
      st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b1 is locally-compact
   holds b2 is locally-compact;

:: YELLOW14:th 29
theorem
for b1, b2 being TopStruct
      st TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#) &
         b1 is compact
   holds b2 is compact;

:: YELLOW14:funcnot 2 => YELLOW14:func 2
definition
  let a1 be non empty set;
  let a2 be non empty TopSpace-like TopStruct;
  let a3 be Element of the carrier of product (a1 --> a2);
  let a4 be Element of a1;
  redefine func a3 . a4 -> Element of the carrier of a2;
end;

:: YELLOW14:th 30
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3, b4 being Element of the carrier of product b2 holds
   b3 in Cl {b4}
iff
   for b5 being Element of b1 holds
      b3 . b5 in Cl {b4 . b5};

:: YELLOW14:th 31
theorem
for b1 being non empty set
for b2 being non empty TopSpace-like TopStruct
for b3, b4 being Element of the carrier of product (b1 --> b2) holds
   b3 in Cl {b4}
iff
   for b5 being Element of b1 holds
      b3 . b5 in Cl {b4 . b5};

:: YELLOW14:th 32
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non-Empty TopSpace-yielding ManySortedSet of b1
for b4 being Element of the carrier of product b3 holds
   pi(Cl {b4},b2) = Cl {b4 . b2};

:: YELLOW14:th 33
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty TopSpace-like TopStruct
for b4 being Element of the carrier of product (b1 --> b3) holds
   pi(Cl {b4},b2) = Cl {b4 . b2};

:: YELLOW14:th 34
theorem
for b1, b2 being non empty TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
      st b3 = id b1 & b4 = id b1 & b3 is continuous(b1, b2) & b4 is continuous(b2, b1)
   holds TopStruct(#the carrier of b1,the topology of b1#) = TopStruct(#the carrier of b2,the topology of b2#);

:: YELLOW14:th 35
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st corestr b3 is continuous(b1, Image b3)
   holds b3 is continuous(b1, b2);

:: YELLOW14:funcreg 4
registration
  let a1 be non empty TopSpace-like TopStruct;
  let a2 be non empty SubSpace of a1;
  cluster incl a2 -> Function-like quasi_total continuous;
end;

:: YELLOW14:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 * b2 = b2
   holds (corestr b2) * incl Image b2 = id Image b2;

:: YELLOW14:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
   corestr incl b2 is being_homeomorphism(b2, Image incl b2);

:: YELLOW14:th 38
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is TopSpace-like discerning TopStruct
   holds product b2 is discerning;

:: YELLOW14:funcreg 5
registration
  let a1 be non empty set;
  let a2 be non empty TopSpace-like discerning TopStruct;
  cluster product (a1 --> a2) -> strict TopSpace-like discerning;
end;

:: YELLOW14:th 39
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
      st for b3 being Element of b1 holds
           b2 . b3 is being_T1 & b2 . b3 is TopSpace-like
   holds product b2 is being_T1;

:: YELLOW14:funcreg 6
registration
  let a1 be non empty set;
  let a2 be non empty TopSpace-like being_T1 TopStruct;
  cluster product (a1 --> a2) -> strict TopSpace-like being_T1;
end;