Article PENCIL_3, MML version 4.99.1005

:: PENCIL_3:th 1
theorem
for b1 being non empty non void TopStruct
for b2 being Function-like quasi_total isomorphic Relation of the carrier of b1,the carrier of b1
for b3, b4 being Element of the carrier of b1
      st b3,b4 are_collinear
   holds b2 . b3,b2 . b4 are_collinear;

:: PENCIL_3:th 2
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being 1-sorted-yielding non-Trivial-yielding ManySortedSet of b1 holds
   b3 . b2 is not trivial;

:: PENCIL_3:th 3
theorem
for b1 being non void identifying_close_blocks TopStruct
      st b1 is strongly_connected
   holds b1 is without_isolated_points;

:: PENCIL_3:th 4
theorem
for b1 being non empty non void identifying_close_blocks TopStruct
      st b1 is strongly_connected
   holds b1 is connected;

:: PENCIL_3:th 5
theorem
for b1 being non void non degenerated TopStruct
for b2 being Element of the topology of b1 holds
   ex b3 being Element of the carrier of b1 st
      not b3 in b2;

:: PENCIL_3:th 6
theorem
for b1 being non empty set
for b2 being non-Empty TopStruct-yielding ManySortedSet of b1
for b3 being Element of the carrier of Segre_Product b2 holds
   b3 is Element of Carrier b2;

:: PENCIL_3:th 7
theorem
for b1 being non empty set
for b2 being 1-sorted-yielding ManySortedSet of b1
for b3 being Element of b1 holds
   (Carrier b2) . b3 = [#] (b2 . b3);

:: PENCIL_3:th 8
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being TopStruct-yielding non-Trivial-yielding ManySortedSet of b1 holds
   ex b4 being non trivial-yielding Segre-like ManySortedSubset of Carrier b3 st
      indx b4 = b2 & product b4 is Segre-Coset of b3;

:: PENCIL_3:th 9
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being TopStruct-yielding non-Trivial-yielding ManySortedSet of b1
for b4 being Element of the carrier of Segre_Product b3 holds
   ex b5 being non trivial-yielding Segre-like ManySortedSubset of Carrier b3 st
      indx b5 = b2 & product b5 is Segre-Coset of b3 & b4 in product b5;

:: PENCIL_3:th 10
theorem
for b1 being non empty set
for b2 being TopStruct-yielding non-Trivial-yielding ManySortedSet of b1
for b3 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
      st product b3 is Segre-Coset of b2
   holds b3 . indx b3 = [#] (b2 . indx b3);

:: PENCIL_3:th 11
theorem
for b1 being non empty set
for b2 being TopStruct-yielding non-Trivial-yielding ManySortedSet of b1
for b3, b4 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
      st product b3 is Segre-Coset of b2 & product b4 is Segre-Coset of b2 & indx b3 = indx b4 & product b3 meets product b4
   holds b3 = b4;

:: PENCIL_3:th 12
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
for b4 being Element of the topology of b2 . indx b3 holds
   product (b3 +*(indx b3,b4)) is Element of the topology of Segre_Product b2;

:: PENCIL_3:th 13
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of the carrier of b2 . b3
for b5 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
      st b3 <> indx b5
   holds b5 +*(b3,{b4}) is non trivial-yielding Segre-like ManySortedSubset of Carrier b2;

:: PENCIL_3:th 14
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of bool the carrier of b2 . b3
for b5 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2 holds
   product (b5 +*(b3,b4)) is Element of bool the carrier of Segre_Product b2;

:: PENCIL_3:th 15
theorem
for b1 being non empty set
for b2 being ManySortedSet of b1
for b3 being Element of b1 holds
   {b2} . b3 is not empty & {b2} . b3 is trivial;

:: PENCIL_3:th 16
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being PLS-yielding ManySortedSet of b1
for b4 being Element of the topology of b3 . b2
for b5 being Element of Carrier b3 holds
   product ({b5} +*(b2,b4)) is Element of the topology of Segre_Product b3;

:: PENCIL_3:th 17
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3, b4 being Element of the carrier of Segre_Product b2
      st b3 <> b4
   holds    b3,b4 are_collinear
   iff
      for b5, b6 being ManySortedSet of b1
            st b5 = b3 & b6 = b4
         holds ex b7 being Element of b1 st
            (for b8, b9 being Element of the carrier of b2 . b7
                   st b8 = b5 . b7 & b9 = b6 . b7
                holds b8 <> b9 & b8,b9 are_collinear) &
             (for b8 being Element of b1
                   st b8 <> b7
                holds b5 . b8 = b6 . b8);

:: PENCIL_3:th 18
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
for b4 being Element of the carrier of b2 . indx b3 holds
   ex b5 being ManySortedSet of b1 st
      b5 in product b3 &
       {b5 +*(indx b3,b4)} = product (b3 +*(indx b3,{b4}));

:: PENCIL_3:funcnot 1 => PENCIL_3:func 1
definition
  let a1 be non empty finite set;
  let a2, a3 be ManySortedSet of a1;
  func diff(A2,A3) -> natural set equals
    Card {b1 where b1 is Element of a1: a2 . b1 <> a3 . b1};
end;

:: PENCIL_3:def 1
theorem
for b1 being non empty finite set
for b2, b3 being ManySortedSet of b1 holds
diff(b2,b3) = Card {b4 where b4 is Element of b1: b2 . b4 <> b3 . b4};

:: PENCIL_3:th 19
theorem
for b1 being non empty finite set
for b2, b3 being ManySortedSet of b1
for b4 being Element of b1
      st b2 . b4 <> b3 . b4
   holds diff(b2,b3) = (diff(b2,b3 +*(b4,b2 . b4))) + 1;

:: PENCIL_3:prednot 1 => PENCIL_3:pred 1
definition
  let a1 be non empty set;
  let a2 be PLS-yielding ManySortedSet of a1;
  let a3, a4 be Segre-Coset of a2;
  pred A3 '||' A4 means
    for b1 being Element of the carrier of Segre_Product a2
          st b1 in a3
       holds ex b2 being Element of the carrier of Segre_Product a2 st
          b2 in a4 & b1,b2 are_collinear;
end;

:: PENCIL_3:dfs 2
definiens
  let a1 be non empty set;
  let a2 be PLS-yielding ManySortedSet of a1;
  let a3, a4 be Segre-Coset of a2;
To prove
     a3 '||' a4
it is sufficient to prove
  thus for b1 being Element of the carrier of Segre_Product a2
          st b1 in a3
       holds ex b2 being Element of the carrier of Segre_Product a2 st
          b2 in a4 & b1,b2 are_collinear;

:: PENCIL_3:def 2
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3, b4 being Segre-Coset of b2 holds
   b3 '||' b4
iff
   for b5 being Element of the carrier of Segre_Product b2
         st b5 in b3
      holds ex b6 being Element of the carrier of Segre_Product b2 st
         b6 in b4 & b5,b6 are_collinear;

:: PENCIL_3:th 20
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3, b4 being Segre-Coset of b2
   st b3 '||' b4
for b5 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2
for b6, b7 being Segre-Coset of b2
      st b6 = b5 .: b3 & b7 = b5 .: b4
   holds b6 '||' b7;

:: PENCIL_3:th 21
theorem
for b1 being non empty set
for b2 being PLS-yielding ManySortedSet of b1
for b3, b4 being Segre-Coset of b2
      st b3 misses b4
   holds    b3 '||' b4
   iff
      for b5, b6 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
            st b3 = product b5 & b4 = product b6
         holds indx b5 = indx b6 &
          (ex b7 being Element of b1 st
             b7 <> indx b5 &
              (for b8 being Element of b1
                    st b8 <> b7
                 holds b5 . b8 = b6 . b8) &
              (for b8, b9 being Element of the carrier of b2 . b7
                    st b5 . b7 = {b8} & b6 . b7 = {b9}
                 holds b8,b9 are_collinear));

:: PENCIL_3:th 22
theorem
for b1 being non empty finite set
for b2 being PLS-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is connected
for b3 being Element of b1
for b4 being Element of the carrier of b2 . b3
for b5, b6 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
      st product b5 is Segre-Coset of b2 & product b6 is Segre-Coset of b2 & b5 = b6 +*(b3,{b4}) & not b4 in b6 . b3
   holds ex b7 being FinSequence of bool the carrier of Segre_Product b2 st
      b7 . 1 = product b5 &
       b7 . len b7 = product b6 &
       (for b8 being natural set
             st b8 in dom b7
          holds b7 . b8 is Segre-Coset of b2) &
       (for b8 being natural set
          st 1 <= b8 & b8 < len b7
       for b9, b10 being Segre-Coset of b2
             st b9 = b7 . b8 & b10 = b7 . (b8 + 1)
          holds b9 misses b10 & b9 '||' b10);

:: PENCIL_3:th 23
theorem
for b1 being non empty finite set
for b2 being PLS-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is connected
for b3, b4 being Segre-Coset of b2
   st b3 misses b4
for b5, b6 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
      st b3 = product b5 & b4 = product b6
   holds    indx b5 = indx b6
   iff
      ex b7 being FinSequence of bool the carrier of Segre_Product b2 st
         b7 . 1 = b3 &
          b7 . len b7 = b4 &
          (for b8 being natural set
                st b8 in dom b7
             holds b7 . b8 is Segre-Coset of b2) &
          (for b8 being natural set
             st 1 <= b8 & b8 < len b7
          for b9, b10 being Segre-Coset of b2
                st b9 = b7 . b8 & b10 = b7 . (b8 + 1)
             holds b9 misses b10 & b9 '||' b10);

:: PENCIL_3:th 24
theorem
for b1 being non empty finite set
for b2 being PLS-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is strongly_connected
for b3 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2
for b4, b5 being Segre-Coset of b2
for b6, b7, b8, b9 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
      st b4 = product b6 & b5 = product b7 & b3 .: b4 = product b8 & b3 .: b5 = product b9 & indx b6 = indx b7
   holds indx b8 = indx b9;

:: PENCIL_3:th 25
theorem
for b1 being non empty finite set
for b2 being PLS-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is strongly_connected
for b3 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2 holds
   ex b4 being Function-like quasi_total bijective Relation of b1,b1 st
      for b5, b6 being Element of b1 holds
         b4 . b5 = b6
      iff
         for b7 being Segre-Coset of b2
         for b8, b9 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
               st b7 = product b8 & b3 .: b7 = product b9 & indx b8 = b5
            holds indx b9 = b6;

:: PENCIL_3:funcnot 2 => PENCIL_3:func 2
definition
  let a1 be non empty finite set;
  let a2 be PLS-yielding ManySortedSet of a1;
  let a3 be Function-like quasi_total isomorphic Relation of the carrier of Segre_Product a2,the carrier of Segre_Product a2;
  assume for b1 being Element of a1 holds
       a2 . b1 is strongly_connected;
  func permutation_of_indices A3 -> Function-like quasi_total bijective Relation of a1,a1 means
    for b1, b2 being Element of a1 holds
       it . b1 = b2
    iff
       for b3 being Segre-Coset of a2
       for b4, b5 being non trivial-yielding Segre-like ManySortedSubset of Carrier a2
             st b3 = product b4 & a3 .: b3 = product b5 & indx b4 = b1
          holds indx b5 = b2;
end;

:: PENCIL_3:def 3
theorem
for b1 being non empty finite set
for b2 being PLS-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is strongly_connected
for b3 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2
for b4 being Function-like quasi_total bijective Relation of b1,b1 holds
      b4 = permutation_of_indices b3
   iff
      for b5, b6 being Element of b1 holds
         b4 . b5 = b6
      iff
         for b7 being Segre-Coset of b2
         for b8, b9 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
               st b7 = product b8 & b3 .: b7 = product b9 & indx b8 = b5
            holds indx b9 = b6;

:: PENCIL_3:funcnot 3 => PENCIL_3:func 3
definition
  let a1 be non empty finite set;
  let a2 be PLS-yielding ManySortedSet of a1;
  let a3 be Function-like quasi_total isomorphic Relation of the carrier of Segre_Product a2,the carrier of Segre_Product a2;
  let a4 be non trivial-yielding Segre-like ManySortedSubset of Carrier a2;
  assume (for b1 being Element of a1 holds
        a2 . b1 is strongly_connected) &
     product a4 is Segre-Coset of a2;
  func canonical_embedding(A3,A4) -> Function-like quasi_total Relation of the carrier of a2 . indx a4,the carrier of a2 . ((permutation_of_indices a3) . indx a4) means
    it is isomorphic(a2 . indx a4, a2 . ((permutation_of_indices a3) . indx a4)) &
     (for b1 being ManySortedSet of a1
        st b1 is Element of the carrier of Segre_Product a2 & b1 in product a4
     for b2 being ManySortedSet of a1
           st b2 = a3 . b1
        holds b2 . ((permutation_of_indices a3) . indx a4) = it . (b1 . indx a4));
end;

:: PENCIL_3:def 4
theorem
for b1 being non empty finite set
for b2 being PLS-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is strongly_connected
for b3 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2
for b4 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
   st product b4 is Segre-Coset of b2
for b5 being Function-like quasi_total Relation of the carrier of b2 . indx b4,the carrier of b2 . ((permutation_of_indices b3) . indx b4) holds
      b5 = canonical_embedding(b3,b4)
   iff
      b5 is isomorphic(b2 . indx b4, b2 . ((permutation_of_indices b3) . indx b4)) &
       (for b6 being ManySortedSet of b1
          st b6 is Element of the carrier of Segre_Product b2 & b6 in product b4
       for b7 being ManySortedSet of b1
             st b7 = b3 . b6
          holds b7 . ((permutation_of_indices b3) . indx b4) = b5 . (b6 . indx b4));

:: PENCIL_3:th 26
theorem
for b1 being non empty finite set
for b2 being PLS-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is strongly_connected
for b3 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2
for b4, b5 being Segre-Coset of b2
   st b4 misses b5 & b4 '||' b5
for b6, b7 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
      st product b6 = b4 & product b7 = b5
   holds canonical_embedding(b3,b6) = canonical_embedding(b3,b7);

:: PENCIL_3:th 27
theorem
for b1 being non empty finite set
for b2 being PLS-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is strongly_connected
for b3 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2
for b4, b5 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
      st product b4 is Segre-Coset of b2 & product b5 is Segre-Coset of b2 & indx b4 = indx b5
   holds canonical_embedding(b3,b4) = canonical_embedding(b3,b5);

:: PENCIL_3:funcnot 4 => PENCIL_3:func 4
definition
  let a1 be non empty finite set;
  let a2 be PLS-yielding ManySortedSet of a1;
  let a3 be Function-like quasi_total isomorphic Relation of the carrier of Segre_Product a2,the carrier of Segre_Product a2;
  let a4 be Element of a1;
  assume for b1 being Element of a1 holds
       a2 . b1 is strongly_connected;
  func canonical_embedding(A3,A4) -> Function-like quasi_total Relation of the carrier of a2 . a4,the carrier of a2 . ((permutation_of_indices a3) . a4) means
    for b1 being non trivial-yielding Segre-like ManySortedSubset of Carrier a2
          st product b1 is Segre-Coset of a2 & indx b1 = a4
       holds it = canonical_embedding(a3,b1);
end;

:: PENCIL_3:def 5
theorem
for b1 being non empty finite set
for b2 being PLS-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is strongly_connected
for b3 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2
for b4 being Element of b1
for b5 being Function-like quasi_total Relation of the carrier of b2 . b4,the carrier of b2 . ((permutation_of_indices b3) . b4) holds
      b5 = canonical_embedding(b3,b4)
   iff
      for b6 being non trivial-yielding Segre-like ManySortedSubset of Carrier b2
            st product b6 is Segre-Coset of b2 & indx b6 = b4
         holds b5 = canonical_embedding(b3,b6);

:: PENCIL_3:th 28
theorem
for b1 being non empty finite set
for b2 being PLS-yielding ManySortedSet of b1
   st for b3 being Element of b1 holds
        b2 . b3 is strongly_connected
for b3 being Function-like quasi_total isomorphic Relation of the carrier of Segre_Product b2,the carrier of Segre_Product b2 holds
   ex b4 being Function-like quasi_total bijective Relation of b1,b1 st
      ex b5 being Function-yielding ManySortedSet of b1 st
         for b6 being Element of b1 holds
            b5 . b6 is Function-like quasi_total Relation of the carrier of b2 . b6,the carrier of b2 . (b4 . b6) &
             (for b7 being Function-like quasi_total Relation of the carrier of b2 . b6,the carrier of b2 . (b4 . b6)
                   st b7 = b5 . b6
                holds b7 is isomorphic(b2 . b6, b2 . (b4 . b6))) &
             (for b7 being Element of the carrier of Segre_Product b2
             for b8 being ManySortedSet of b1
                st b8 = b7
             for b9 being ManySortedSet of b1
                st b9 = b3 . b7
             for b10 being Function-like quasi_total Relation of the carrier of b2 . b6,the carrier of b2 . (b4 . b6)
                   st b10 = b5 . b6
                holds b9 . (b4 . b6) = b10 . (b8 . b6));