Article TSEP_2, MML version 4.99.1005

:: TSEP_2:th 1
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
b2 ` \ (b3 `) = b3 \ b2;

:: TSEP_2:prednot 1 => TSEP_2:pred 1
definition
  let a1 be TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  pred A2,A3 constitute_a_decomposition means
    a2 misses a3 & a2 \/ a3 = the carrier of a1;
  symmetry;
::  for a1 being TopSpace-like TopStruct
::  for a2, a3 being Element of bool the carrier of a1
::        st a2,a3 constitute_a_decomposition
::     holds a3,a2 constitute_a_decomposition;
end;

:: TSEP_2:dfs 1
definiens
  let a1 be TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
To prove
     a2,a3 constitute_a_decomposition
it is sufficient to prove
  thus a2 misses a3 & a2 \/ a3 = the carrier of a1;

:: TSEP_2:def 1
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 constitute_a_decomposition
iff
   b2 misses b3 & b2 \/ b3 = the carrier of b1;

:: TSEP_2:prednot 2 => not TSEP_2:pred 1
notation
  let a1 be TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  antonym a2,a3 do_not_constitute_a_decomposition for a2,a3 constitute_a_decomposition;
end;

:: TSEP_2:th 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 constitute_a_decomposition
iff
   b2 misses b3 & b2 \/ b3 = [#] b1;

:: TSEP_2:th 4
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2,b3 constitute_a_decomposition
   holds b2 = b3 ` & b3 = b2 `;

:: TSEP_2:th 5
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st (b2 = b3 ` or b3 = b2 `)
   holds b2,b3 constitute_a_decomposition;

:: TSEP_2:th 6
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   b2,b2 ` constitute_a_decomposition;

:: TSEP_2:th 7
theorem
for b1 being non empty TopSpace-like TopStruct holds
   {} b1,[#] b1 constitute_a_decomposition;

:: TSEP_2:th 8
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   b2,b2 do_not_constitute_a_decomposition;

:: TSEP_2:prednot 3 => TSEP_2:pred 2
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  redefine pred a2,a3 constitute_a_decomposition;
  symmetry;
::  for a1 being non empty TopSpace-like TopStruct
::  for a2, a3 being Element of bool the carrier of a1
::        st a2,a3 constitute_a_decomposition
::     holds a3,a2 constitute_a_decomposition;
  irreflexivity;
::  for a1 being non empty TopSpace-like TopStruct
::  for a2 being Element of bool the carrier of a1 holds
::     not a2,a2 constitute_a_decomposition;
end;

:: TSEP_2:th 9
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2,b3 constitute_a_decomposition & b3,b4 constitute_a_decomposition
   holds b2 = b4;

:: TSEP_2:th 10
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2,b3 constitute_a_decomposition
   holds Cl b2,Int b3 constitute_a_decomposition & Int b2,Cl b3 constitute_a_decomposition;

:: TSEP_2:th 11
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Element of bool the carrier of b1 holds
   Cl b2,Int (b2 `) constitute_a_decomposition & Cl (b2 `),Int b2 constitute_a_decomposition & Int b2,Cl (b2 `) constitute_a_decomposition & Int (b2 `),Cl b2 constitute_a_decomposition;

:: TSEP_2:th 12
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2,b3 constitute_a_decomposition
   holds    b2 is open(b1)
   iff
      b3 is closed(b1);

:: TSEP_2:th 13
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2,b3 constitute_a_decomposition
   holds    b2 is closed(b1)
   iff
      b3 is open(b1);

:: TSEP_2:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b2,b3 constitute_a_decomposition & b4,b5 constitute_a_decomposition
   holds b2 /\ b4,b3 \/ b5 constitute_a_decomposition;

:: TSEP_2:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b2,b3 constitute_a_decomposition & b4,b5 constitute_a_decomposition
   holds b2 \/ b4,b3 /\ b5 constitute_a_decomposition;

:: TSEP_2:th 16
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition
   holds    b2,b3 are_weakly_separated
   iff
      b4,b5 are_weakly_separated;

:: TSEP_2:th 17
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_weakly_separated
iff
   b2 `,b3 ` are_weakly_separated;

:: TSEP_2:th 18
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b2,b3 are_separated
   holds b4,b5 are_weakly_separated;

:: TSEP_2:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b2 misses b3 & b4,b5 are_weakly_separated
   holds b2,b3 are_separated;

:: TSEP_2:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b4 \/ b5 = the carrier of b1 & b4,b5 are_weakly_separated
   holds b2,b3 are_separated;

:: TSEP_2:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2,b3 constitute_a_decomposition
   holds    b2,b3 are_weakly_separated
   iff
      b2,b3 are_separated;

:: TSEP_2:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_weakly_separated
iff
   (b2 \/ b3) \ b2,(b2 \/ b3) \ b3 are_separated;

:: TSEP_2:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b4 c= b2 & b5 c= b3 & b4 \/ b5 = b2 \/ b3 & b4,b5 are_weakly_separated
   holds b2,b3 are_weakly_separated;

:: TSEP_2:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_weakly_separated
iff
   b2 \ (b2 /\ b3),b3 \ (b2 /\ b3) are_separated;

:: TSEP_2:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b4 c= b2 & b5 c= b3 & b4 /\ b5 = b2 /\ b3 & b2,b3 are_weakly_separated
   holds b4,b5 are_weakly_separated;

:: TSEP_2:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being non empty SubSpace of b1
for b5, b6 being Element of bool the carrier of b4
      st b5 = b2 & b6 = b3
   holds    b2,b3 are_separated
   iff
      b5,b6 are_separated;

:: TSEP_2:th 27
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being non empty SubSpace of b1
for b5, b6 being Element of bool the carrier of b4
      st b5 = (the carrier of b4) /\ b2 & b6 = (the carrier of b4) /\ b3 & b2,b3 are_separated
   holds b5,b6 are_separated;

:: TSEP_2:th 28
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being non empty SubSpace of b1
for b5, b6 being Element of bool the carrier of b4
      st b5 = b2 & b6 = b3
   holds    b2,b3 are_weakly_separated
   iff
      b5,b6 are_weakly_separated;

:: TSEP_2:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being non empty SubSpace of b1
for b5, b6 being Element of bool the carrier of b4
      st b5 = (the carrier of b4) /\ b2 & b6 = (the carrier of b4) /\ b3 & b2,b3 are_weakly_separated
   holds b5,b6 are_weakly_separated;

:: TSEP_2:prednot 4 => TSEP_2:pred 3
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be SubSpace of a1;
  pred A2,A3 constitute_a_decomposition means
    for b1, b2 being Element of bool the carrier of a1
          st b1 = the carrier of a2 & b2 = the carrier of a3
       holds b1,b2 constitute_a_decomposition;
  symmetry;
::  for a1 being non empty TopSpace-like TopStruct
::  for a2, a3 being SubSpace of a1
::        st a2,a3 constitute_a_decomposition
::     holds a3,a2 constitute_a_decomposition;
end;

:: TSEP_2:dfs 2
definiens
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be SubSpace of a1;
To prove
     a2,a3 constitute_a_decomposition
it is sufficient to prove
  thus for b1, b2 being Element of bool the carrier of a1
          st b1 = the carrier of a2 & b2 = the carrier of a3
       holds b1,b2 constitute_a_decomposition;

:: TSEP_2:def 2
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being SubSpace of b1 holds
   b2,b3 constitute_a_decomposition
iff
   for b4, b5 being Element of bool the carrier of b1
         st b4 = the carrier of b2 & b5 = the carrier of b3
      holds b4,b5 constitute_a_decomposition;

:: TSEP_2:prednot 5 => not TSEP_2:pred 3
notation
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be SubSpace of a1;
  antonym a2,a3 do_not_constitute_a_decomposition for a2,a3 constitute_a_decomposition;
end;

:: TSEP_2:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
   b2,b3 constitute_a_decomposition
iff
   b2 misses b3 &
    TopStruct(#the carrier of b1,the topology of b1#) = b2 union b3;

:: TSEP_2:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1 holds
   b2,b2 do_not_constitute_a_decomposition;

:: TSEP_2:prednot 6 => TSEP_2:pred 4
definition
  let a1 be non empty TopSpace-like TopStruct;
  let a2, a3 be non empty SubSpace of a1;
  redefine pred a2,a3 constitute_a_decomposition;
  symmetry;
::  for a1 being non empty TopSpace-like TopStruct
::  for a2, a3 being non empty SubSpace of a1
::        st a2,a3 constitute_a_decomposition
::     holds a3,a2 constitute_a_decomposition;
  irreflexivity;
::  for a1 being non empty TopSpace-like TopStruct
::  for a2 being non empty SubSpace of a1 holds
::     not a2,a2 constitute_a_decomposition;
end;

:: TSEP_2:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
      st b2,b3 constitute_a_decomposition & b3,b4 constitute_a_decomposition
   holds TopStruct(#the carrier of b2,the topology of b2#) = TopStruct(#the carrier of b4,the topology of b4#);

:: TSEP_2:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
      st b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition
   holds    b4 union b5 = TopStruct(#the carrier of b1,the topology of b1#)
   iff
      b2 misses b3;

:: TSEP_2:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2,b3 constitute_a_decomposition
   holds    b2 is open(b1)
   iff
      b3 is closed(b1);

:: TSEP_2:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2,b3 constitute_a_decomposition
   holds    b2 is closed(b1)
   iff
      b3 is open(b1);

:: TSEP_2:th 37
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
      st b2 meets b3 & b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition
   holds b2 meet b3,b4 union b5 constitute_a_decomposition;

:: TSEP_2:th 38
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
      st b2 meets b3 & b4,b2 constitute_a_decomposition & b5,b3 constitute_a_decomposition
   holds b4 union b5,b2 meet b3 constitute_a_decomposition;

:: TSEP_2:th 39
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being SubSpace of b1
      st b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b2,b3 are_weakly_separated
   holds b4,b5 are_weakly_separated;

:: TSEP_2:th 40
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
      st b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b2,b3 are_separated
   holds b4,b5 are_weakly_separated;

:: TSEP_2:th 41
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
      st b2,b4 constitute_a_decomposition & b3,b5 constitute_a_decomposition & b2 misses b3 & b4,b5 are_weakly_separated
   holds b2,b3 are_separated;

:: TSEP_2:th 42
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
      st b2,b4 constitute_a_decomposition &
         b3,b5 constitute_a_decomposition &
         b4 union b5 = TopStruct(#the carrier of b1,the topology of b1#) &
         b4,b5 are_weakly_separated
   holds b2,b3 are_separated;

:: TSEP_2:th 43
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2,b3 constitute_a_decomposition
   holds    b2,b3 are_weakly_separated
   iff
      b2,b3 are_separated;

:: TSEP_2:th 44
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
      st b4 is SubSpace of b2 & b5 is SubSpace of b3 & b4 union b5 = b2 union b3 & b4,b5 are_weakly_separated
   holds b2,b3 are_weakly_separated;

:: TSEP_2:th 45
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4, b5 being non empty SubSpace of b1
      st b4 is SubSpace of b2 & b5 is SubSpace of b3 & b4 meets b5 & b4 meet b5 = b2 meet b3 & b2,b3 are_weakly_separated
   holds b4,b5 are_weakly_separated;

:: TSEP_2:th 46
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3, b4 being SubSpace of b1
for b5, b6 being SubSpace of b2
      st the carrier of b3 = the carrier of b5 & the carrier of b4 = the carrier of b6
   holds    b3,b4 are_separated
   iff
      b5,b6 are_separated;

:: TSEP_2:th 47
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
   st b3 meets b2 & b4 meets b2
for b5, b6 being SubSpace of b2
      st b5 = b3 meet b2 & b6 = b4 meet b2 & b3,b4 are_separated
   holds b5,b6 are_separated;

:: TSEP_2:th 48
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty SubSpace of b1
for b3, b4 being SubSpace of b1
for b5, b6 being SubSpace of b2
      st the carrier of b3 = the carrier of b5 & the carrier of b4 = the carrier of b6
   holds    b3,b4 are_weakly_separated
   iff
      b5,b6 are_weakly_separated;

:: TSEP_2:th 49
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
   st b3 meets b2 & b4 meets b2
for b5, b6 being SubSpace of b2
      st b5 = b3 meet b2 & b6 = b4 meet b2 & b3,b4 are_weakly_separated
   holds b5,b6 are_weakly_separated;