Article YELLOW17, MML version 4.99.1005

:: YELLOW17:th 1
theorem
for b1 being Relation-like Function-like set
for b2, b3 being set
for b4 being Element of bool (b1 . b2)
      st (proj(b1,b2)) " {b3} meets (proj(b1,b2)) " b4
   holds b3 in b4;

:: YELLOW17:th 2
theorem
for b1, b2 being Relation-like Function-like set
for b3, b4 being set
      st b4 in b1 . b3 & b2 in product b1
   holds b2 +*(b3,b4) in product b1;

:: YELLOW17:th 3
theorem
for b1 being Relation-like Function-like set
for b2 being set
      st b2 in proj1 b1 & product b1 <> {}
   holds proj2 proj(b1,b2) = b1 . b2;

:: YELLOW17:th 4
theorem
for b1 being Relation-like Function-like set
for b2 being set
      st b2 in proj1 b1
   holds (proj(b1,b2)) " (b1 . b2) = product b1;

:: YELLOW17:th 5
theorem
for b1, b2 being Relation-like Function-like set
for b3, b4 being set
      st b4 in b1 . b3 & b3 in proj1 b1 & b2 in product b1
   holds b2 +*(b3,b4) in (proj(b1,b3)) " {b4};

:: YELLOW17:th 6
theorem
for b1, b2 being Relation-like Function-like set
for b3, b4, b5 being set
for b6 being Element of bool (b1 . b4)
      st b5 in b1 . b3 & b3 in proj1 b1 & b2 in product b1 & b3 <> b4
   holds    b2 in (proj(b1,b4)) " b6
   iff
      b2 +*(b3,b5) in (proj(b1,b4)) " b6;

:: YELLOW17:th 7
theorem
for b1 being Relation-like Function-like set
for b2, b3, b4 being set
for b5 being Element of bool (b1 . b3)
      st product b1 <> {} & b4 in b1 . b2 & b2 in proj1 b1 & b3 in proj1 b1 & b5 <> b1 . b3
   holds    (proj(b1,b2)) " {b4} c= (proj(b1,b3)) " b5
   iff
      b2 = b3 & b4 in b5;

:: YELLOW17:sch 1
scheme YELLOW17:sch 1
{F1 -> non empty set,
  F2 -> non-Empty TopSpace-yielding ManySortedSet of F1()}:
ex b1 being Element of the carrier of product F2() st
   for b2 being Element of F1() holds
      P1[b1 . b2, b2]
provided
   for b1 being Element of F1() holds
      ex b2 being Element of the carrier of F2() . b1 st
         P1[b2, b1];


:: YELLOW17:th 8
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of the carrier of product b2 holds
   (proj(b2,b3)) . b4 = b4 . b3;

:: YELLOW17:th 9
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of the carrier of b2 . b3
for b5 being Element of bool the carrier of b2 . b3
      st (proj(b2,b3)) " {b4} meets (proj(b2,b3)) " b5
   holds b4 in b5;

:: YELLOW17:th 10
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1 holds
   (proj(b2,b3)) " [#] (b2 . b3) = [#] product b2;

:: YELLOW17:th 11
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of the carrier of b2 . b3
for b5 being Element of the carrier of product b2 holds
   b5 +*(b3,b4) in (proj(b2,b3)) " {b4};

:: YELLOW17:th 12
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3, b4 being Element of b1
for b5 being Element of the carrier of b2 . b3
for b6 being Element of bool the carrier of b2 . b4
      st b6 <> [#] (b2 . b4)
   holds    (proj(b2,b3)) " {b5} c= (proj(b2,b4)) " b6
   iff
      b3 = b4 & b5 in b6;

:: YELLOW17:th 13
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3, b4 being Element of b1
for b5 being Element of the carrier of b2 . b3
for b6 being Element of bool the carrier of b2 . b4
for b7 being Element of the carrier of product b2
      st b3 <> b4
   holds    b7 in (proj(b2,b4)) " b6
   iff
      b7 +*(b3,b5) in (proj(b2,b4)) " b6;

:: YELLOW17:th 15
theorem
for b1 being non empty TopStruct holds
      b1 is compact
   iff
      for b2 being Element of bool bool the carrier of b1
            st b2 is open(b1) & [#] b1 c= union b2
         holds ex b3 being Element of bool bool the carrier of b1 st
            b3 c= b2 & [#] b1 c= union b3 & b3 is finite;

:: YELLOW17:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being prebasis of b1 holds
      b1 is compact
   iff
      for b3 being Element of bool b2
            st [#] b1 c= union b3
         holds ex b4 being finite Element of bool b3 st
            [#] b1 c= union b4;

:: YELLOW17:th 17
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being set
      st b3 in product_prebasis b2
   holds ex b4 being Element of b1 st
      ex b5 being Element of bool the carrier of b2 . b4 st
         b5 is open(b2 . b4) & (proj(b2,b4)) " b5 = b3;

:: YELLOW17:th 18
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of the carrier of b2 . b3
for b5 being set
      st b5 in product_prebasis b2 & (proj(b2,b3)) " {b4} c= b5 & b5 <> [#] product b2
   holds ex b6 being Element of bool the carrier of b2 . b3 st
      b6 <> [#] (b2 . b3) & b4 in b6 & b6 is open(b2 . b3) & b5 = (proj(b2,b3)) " b6;

:: YELLOW17:th 19
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being non empty Element of bool bool the carrier of b2 . b3
      st [#] (b2 . b3) c= union b4
   holds [#] product b2 c= union {(proj(b2,b3)) " b5 where b5 is Element of b4: TRUE};

:: YELLOW17:th 20
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of the carrier of b2 . b3
for b5 being Element of bool product_prebasis b2
      st (proj(b2,b3)) " {b4} c= union b5 &
         (for b6 being set
               st b6 in product_prebasis b2 & b6 in b5
            holds not (proj(b2,b3)) " {b4} c= b6)
   holds [#] product b2 c= union b5;

:: YELLOW17:th 21
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of bool product_prebasis b2
   st for b5 being finite Element of bool b4 holds
        not [#] product b2 c= union b5
for b5 being Element of the carrier of b2 . b3
for b6 being finite Element of bool b4
      st (proj(b2,b3)) " {b5} c= union b6
   holds ex b7 being set st
      b7 in product_prebasis b2 & b7 in b6 & (proj(b2,b3)) " {b5} c= b7;

:: YELLOW17:th 22
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of bool product_prebasis b2
   st for b5 being finite Element of bool b4 holds
        not [#] product b2 c= union b5
for b5 being Element of the carrier of b2 . b3
for b6 being finite Element of bool b4
      st (proj(b2,b3)) " {b5} c= union b6
   holds ex b7 being Element of bool the carrier of b2 . b3 st
      b7 <> [#] (b2 . b3) & b5 in b7 & (proj(b2,b3)) " b7 in b6 & b7 is open(b2 . b3);

:: YELLOW17:th 23
theorem
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3 being Element of b1
for b4 being Element of bool product_prebasis b2
      st (for b5 being Element of b1 holds
            b2 . b5 is compact) &
         (for b5 being finite Element of bool b4 holds
            not [#] product b2 c= union b5)
   holds ex b5 being Element of the carrier of b2 . b3 st
      for b6 being finite Element of bool b4 holds
         not (proj(b2,b3)) " {b5} c= union b6;

:: YELLOW17:th 24
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 compact
   holds product b2 is compact;