Article TSEP_1, MML version 4.99.1005

:: TSEP_1:th 1
theorem
for b1 being TopStruct
for b2 being SubSpace of b1 holds
   the carrier of b2 is Element of bool the carrier of b1;

:: TSEP_1:th 2
theorem
for b1 being TopStruct holds
   b1 is SubSpace of b1;

:: TSEP_1:th 3
theorem
for b1 being strict TopStruct holds
   b1 | [#] b1 = b1;

:: TSEP_1:th 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being SubSpace of b1 holds
   the carrier of b2 c= the carrier of b3
iff
   b2 is SubSpace of b3;

:: TSEP_1:th 5
theorem
for b1 being TopStruct
for b2, b3 being SubSpace of b1
      st the carrier of b2 = the carrier of b3
   holds TopStruct(#the carrier of b2,the topology of b2#) = TopStruct(#the carrier of b3,the topology of b3#);

:: TSEP_1:th 6
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being SubSpace of b1
      st b2 is SubSpace of b3 & b3 is SubSpace of b2
   holds TopStruct(#the carrier of b2,the topology of b2#) = TopStruct(#the carrier of b3,the topology of b3#);

:: TSEP_1:th 7
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being SubSpace of b2 holds
   b3 is SubSpace of b1;

:: TSEP_1:th 8
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b3 is closed(b1) & b3 c= the carrier of b2 & b4 c= b3 & b4 = b5
   holds    b5 is closed(b2)
   iff
      b4 is closed(b1);

:: TSEP_1:th 9
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3, b4 being Element of bool the carrier of b1
for b5 being Element of bool the carrier of b2
      st b3 is open(b1) & b3 c= the carrier of b2 & b4 c= b3 & b4 = b5
   holds    b5 is open(b2)
   iff
      b4 is open(b1);

:: TSEP_1:th 10
theorem
for b1 being non empty TopStruct
for b2 being non empty Element of bool the carrier of b1 holds
   ex b3 being non empty strict SubSpace of b1 st
      b2 = the carrier of b3;

:: TSEP_1:th 11
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
      st b3 = the carrier of b2
   holds    b2 is closed SubSpace of b1
   iff
      b3 is closed(b1);

:: TSEP_1:th 12
theorem
for b1 being TopSpace-like TopStruct
for b2 being closed SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds    b4 is closed(b2)
   iff
      b3 is closed(b1);

:: TSEP_1:th 13
theorem
for b1 being TopSpace-like TopStruct
for b2 being closed SubSpace of b1
for b3 being closed SubSpace of b2 holds
   b3 is closed SubSpace of b1;

:: TSEP_1:th 14
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty closed SubSpace of b1
for b3 being non empty SubSpace of b1
      st the carrier of b2 c= the carrier of b3
   holds b2 is non empty closed SubSpace of b3;

:: TSEP_1:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
      st b2 is closed(b1)
   holds ex b3 being non empty strict closed SubSpace of b1 st
      b2 = the carrier of b3;

:: TSEP_1:attrnot 1 => TSEP_1:attr 1
definition
  let a1 be TopStruct;
  let a2 be SubSpace of a1;
  attr a2 is open means
    for b1 being Element of bool the carrier of a1
          st b1 = the carrier of a2
       holds b1 is open(a1);
end;

:: TSEP_1:dfs 1
definiens
  let a1 be TopStruct;
  let a2 be SubSpace of a1;
To prove
     a2 is open
it is sufficient to prove
  thus for b1 being Element of bool the carrier of a1
          st b1 = the carrier of a2
       holds b1 is open(a1);

:: TSEP_1:def 1
theorem
for b1 being TopStruct
for b2 being SubSpace of b1 holds
      b2 is open(b1)
   iff
      for b3 being Element of bool the carrier of b1
            st b3 = the carrier of b2
         holds b3 is open(b1);

:: TSEP_1:exreg 1
registration
  let a1 be TopSpace-like TopStruct;
  cluster strict TopSpace-like open SubSpace of a1;
end;

:: TSEP_1:exreg 2
registration
  let a1 be non empty TopSpace-like TopStruct;
  cluster non empty strict TopSpace-like open SubSpace of a1;
end;

:: TSEP_1:th 16
theorem
for b1 being TopSpace-like TopStruct
for b2 being SubSpace of b1
for b3 being Element of bool the carrier of b1
      st b3 = the carrier of b2
   holds    b2 is open SubSpace of b1
   iff
      b3 is open(b1);

:: TSEP_1:th 17
theorem
for b1 being TopSpace-like TopStruct
for b2 being open SubSpace of b1
for b3 being Element of bool the carrier of b1
for b4 being Element of bool the carrier of b2
      st b3 = b4
   holds    b4 is open(b2)
   iff
      b3 is open(b1);

:: TSEP_1:th 18
theorem
for b1 being TopSpace-like TopStruct
for b2 being open SubSpace of b1
for b3 being open SubSpace of b2 holds
   b3 is open SubSpace of b1;

:: TSEP_1:th 19
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being open SubSpace of b1
for b3 being non empty SubSpace of b1
      st the carrier of b2 c= the carrier of b3
   holds b2 is open SubSpace of b3;

:: TSEP_1:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Element of bool the carrier of b1
      st b2 is open(b1)
   holds ex b3 being non empty strict open SubSpace of b1 st
      b2 = the carrier of b3;

:: TSEP_1:funcnot 1 => TSEP_1:func 1
definition
  let a1 be non empty TopStruct;
  let a2, a3 be non empty SubSpace of a1;
  func A2 union A3 -> non empty strict SubSpace of a1 means
    the carrier of it = (the carrier of a2) \/ the carrier of a3;
  commutativity;
::  for a1 being non empty TopStruct
::  for a2, a3 being non empty SubSpace of a1 holds
::  a2 union a3 = a3 union a2;
end;

:: TSEP_1:def 2
theorem
for b1 being non empty TopStruct
for b2, b3 being non empty SubSpace of b1
for b4 being non empty strict SubSpace of b1 holds
      b4 = b2 union b3
   iff
      the carrier of b4 = (the carrier of b2) \/ the carrier of b3;

:: TSEP_1:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1 holds
(b2 union b3) union b4 = b2 union (b3 union b4);

:: TSEP_1:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
b2 is SubSpace of b2 union b3;

:: TSEP_1:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
   b2 is SubSpace of b3
iff
   b2 union b3 = TopStruct(#the carrier of b3,the topology of b3#);

:: TSEP_1:th 24
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty closed SubSpace of b1 holds
b2 union b3 is closed SubSpace of b1;

:: TSEP_1:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty open SubSpace of b1 holds
b2 union b3 is open SubSpace of b1;

:: TSEP_1:prednot 1 => TSEP_1:pred 1
definition
  let a1, a2 be 1-sorted;
  pred A1 misses A2 means
    the carrier of a1 misses the carrier of a2;
  symmetry;
::  for a1, a2 being 1-sorted
::        st a1 misses a2
::     holds a2 misses a1;
end;

:: TSEP_1:dfs 3
definiens
  let a1, a2 be 1-sorted;
To prove
     a1 misses a2
it is sufficient to prove
  thus the carrier of a1 misses the carrier of a2;

:: TSEP_1:def 3
theorem
for b1, b2 being 1-sorted holds
   b1 misses b2
iff
   the carrier of b1 misses the carrier of b2;

:: TSEP_1:prednot 2 => not TSEP_1:pred 1
notation
  let a1, a2 be 1-sorted;
  antonym a1 meets a2 for a1 misses a2;
end;

:: TSEP_1:funcnot 2 => TSEP_1:func 2
definition
  let a1 be non empty TopStruct;
  let a2, a3 be non empty SubSpace of a1;
  assume a2 meets a3;
  func A2 meet A3 -> non empty strict SubSpace of a1 means
    the carrier of it = (the carrier of a2) /\ the carrier of a3;
end;

:: TSEP_1:def 5
theorem
for b1 being non empty TopStruct
for b2, b3 being non empty SubSpace of b1
   st b2 meets b3
for b4 being non empty strict SubSpace of b1 holds
      b4 = b2 meet b3
   iff
      the carrier of b4 = (the carrier of b2) /\ the carrier of b3;

:: TSEP_1:th 29
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1 holds
(b2 misses b3 or b2 meet b3 = b3 meet b2) &
 ((b2 misses b3 or b2 meet b3 misses b4) & (b3 misses b4 or b2 misses b3 meet b4) or (b2 meet b3) meet b4 = b2 meet (b3 meet b4));

:: TSEP_1:th 30
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2 meets b3
   holds b2 meet b3 is SubSpace of b2 & b2 meet b3 is SubSpace of b3;

:: TSEP_1:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2 meets b3
   holds (b2 is SubSpace of b3 implies b2 meet b3 = TopStruct(#the carrier of b2,the topology of b2#)) &
    (b2 meet b3 = TopStruct(#the carrier of b2,the topology of b2#) implies b2 is SubSpace of b3) &
    (b3 is SubSpace of b2 implies b2 meet b3 = TopStruct(#the carrier of b3,the topology of b3#)) &
    (b2 meet b3 = TopStruct(#the carrier of b3,the topology of b3#) implies b3 is SubSpace of b2);

:: TSEP_1:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty closed SubSpace of b1
      st b2 meets b3
   holds b2 meet b3 is closed SubSpace of b1;

:: TSEP_1:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty open SubSpace of b1
      st b2 meets b3
   holds b2 meet b3 is open SubSpace of b1;

:: TSEP_1:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2 meets b3
   holds b2 meet b3 is SubSpace of b2 union b3;

:: TSEP_1:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
      st b2 meets b4 & b4 meets b3
   holds (b2 union b3) meet b4 = (b2 meet b4) union (b3 meet b4) &
    b4 meet (b2 union b3) = (b4 meet b2) union (b4 meet b3);

:: TSEP_1:th 36
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
      st b2 meets b3
   holds (b2 meet b3) union b4 = (b2 union b4) meet (b3 union b4) &
    b4 union (b2 meet b3) = (b4 union b2) meet (b4 union b3);

:: TSEP_1:prednot 3 => not CONNSP_1:pred 1
notation
  let a1 be TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  antonym a2,a3 are_not_separated for a2,a3 are_separated;
end;

:: TSEP_1:th 38
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is closed(b1) & b3 is closed(b1)
   holds    b2 misses b3
   iff
      b2,b3 are_separated;

:: TSEP_1:th 39
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 \/ b3 is closed(b1) & b2,b3 are_separated
   holds b2 is closed(b1) & b3 is closed(b1);

:: TSEP_1:th 40
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 misses b3 & b2 is open(b1)
   holds b2 misses Cl b3;

:: TSEP_1:th 41
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open(b1) & b3 is open(b1)
   holds    b2 misses b3
   iff
      b2,b3 are_separated;

:: TSEP_1:th 42
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 \/ b3 is open(b1) & b2,b3 are_separated
   holds b2 is open(b1) & b3 is open(b1);

:: TSEP_1:th 43
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2,b3 are_separated
   holds b2 /\ b4,b3 /\ b4 are_separated;

:: TSEP_1:th 44
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st (b2,b4 are_separated or b3,b4 are_separated)
   holds b2 /\ b3,b4 are_separated;

:: TSEP_1:th 45
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1 holds
   b2,b4 are_separated & b3,b4 are_separated
iff
   b2 \/ b3,b4 are_separated;

:: TSEP_1:th 46
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_separated
iff
   ex b4, b5 being Element of bool the carrier of b1 st
      b2 c= b4 & b3 c= b5 & b4 misses b3 & b5 misses b2 & b4 is closed(b1) & b5 is closed(b1);

:: TSEP_1:th 47
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_separated
iff
   ex b4, b5 being Element of bool the carrier of b1 st
      b2 c= b4 & b3 c= b5 & b4 /\ b5 misses b2 \/ b3 & b4 is closed(b1) & b5 is closed(b1);

:: TSEP_1:th 48
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_separated
iff
   ex b4, b5 being Element of bool the carrier of b1 st
      b2 c= b4 & b3 c= b5 & b4 misses b3 & b5 misses b2 & b4 is open(b1) & b5 is open(b1);

:: TSEP_1:th 49
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_separated
iff
   ex b4, b5 being Element of bool the carrier of b1 st
      b2 c= b4 & b3 c= b5 & b4 /\ b5 misses b2 \/ b3 & b4 is open(b1) & b5 is open(b1);

:: TSEP_1:prednot 4 => TSEP_1:pred 2
definition
  let a1 be TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  pred A2,A3 are_weakly_separated means
    a2 \ a3,a3 \ a2 are_separated;
  symmetry;
::  for a1 being TopStruct
::  for a2, a3 being Element of bool the carrier of a1
::        st a2,a3 are_weakly_separated
::     holds a3,a2 are_weakly_separated;
end;

:: TSEP_1:dfs 5
definiens
  let a1 be TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
To prove
     a2,a3 are_weakly_separated
it is sufficient to prove
  thus a2 \ a3,a3 \ a2 are_separated;

:: TSEP_1:def 7
theorem
for b1 being TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_weakly_separated
iff
   b2 \ b3,b3 \ b2 are_separated;

:: TSEP_1:prednot 5 => not TSEP_1:pred 2
notation
  let a1 be TopStruct;
  let a2, a3 be Element of bool the carrier of a1;
  antonym a2,a3 are_not_weakly_separated for a2,a3 are_weakly_separated;
end;

:: TSEP_1:th 51
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2 misses b3 & b2,b3 are_weakly_separated
iff
   b2,b3 are_separated;

:: TSEP_1:th 52
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 c= b3
   holds b2,b3 are_weakly_separated;

:: TSEP_1:th 53
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is closed(b1) & b3 is closed(b1)
   holds b2,b3 are_weakly_separated;

:: TSEP_1:th 54
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 is open(b1) & b3 is open(b1)
   holds b2,b3 are_weakly_separated;

:: TSEP_1:th 55
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2,b3 are_weakly_separated
   holds b2 \/ b4,b3 \/ b4 are_weakly_separated;

:: TSEP_1:th 56
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4, b5 being Element of bool the carrier of b1
      st b4 c= b2 & b5 c= b3 & b3,b2 are_weakly_separated
   holds b3 \/ b4,b2 \/ b5 are_weakly_separated;

:: TSEP_1:th 57
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2,b4 are_weakly_separated & b3,b4 are_weakly_separated
   holds b2 /\ b3,b4 are_weakly_separated;

:: TSEP_1:th 58
theorem
for b1 being TopSpace-like TopStruct
for b2, b3, b4 being Element of bool the carrier of b1
      st b2,b4 are_weakly_separated & b3,b4 are_weakly_separated
   holds b2 \/ b3,b4 are_weakly_separated;

:: TSEP_1:th 59
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_weakly_separated
iff
   ex b4, b5, b6 being Element of bool the carrier of b1 st
      b4 /\ (b2 \/ b3) c= b2 & b5 /\ (b2 \/ b3) c= b3 & b6 /\ (b2 \/ b3) c= b2 /\ b3 & the carrier of b1 = (b4 \/ b5) \/ b6 & b4 is closed(b1) & b5 is closed(b1) & b6 is open(b1);

:: TSEP_1:th 60
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2,b3 are_weakly_separated & not b2 c= b3 & not b3 c= b2
   holds ex b4, b5 being non empty Element of bool the carrier of b1 st
      b4 is closed(b1) &
       b5 is closed(b1) &
       b4 /\ (b2 \/ b3) c= b2 &
       b5 /\ (b2 \/ b3) c= b3 &
       (not b2 \/ b3 c= b4 \/ b5 implies ex b6 being non empty Element of bool the carrier of b1 st
          b6 is open(b1) & b6 /\ (b2 \/ b3) c= b2 /\ b3 & the carrier of b1 = (b4 \/ b5) \/ b6);

:: TSEP_1:th 61
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 \/ b3 = the carrier of b1
   holds    b2,b3 are_weakly_separated
   iff
      ex b4, b5, b6 being Element of bool the carrier of b1 st
         b2 \/ b3 = (b4 \/ b5) \/ b6 & b4 c= b2 & b5 c= b3 & b6 c= b2 /\ b3 & b4 is closed(b1) & b5 is closed(b1) & b6 is open(b1);

:: TSEP_1:th 62
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 \/ b3 = the carrier of b1 & b2,b3 are_weakly_separated & not b2 c= b3 & not b3 c= b2
   holds ex b4, b5 being non empty Element of bool the carrier of b1 st
      b4 is closed(b1) &
       b5 is closed(b1) &
       b4 c= b2 &
       b5 c= b3 &
       (b2 \/ b3 <> b4 \/ b5 implies ex b6 being non empty Element of bool the carrier of b1 st
          b2 \/ b3 = (b4 \/ b5) \/ b6 & b6 c= b2 /\ b3 & b6 is open(b1));

:: TSEP_1:th 63
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
   ex b4, b5, b6 being Element of bool the carrier of b1 st
      b4 /\ (b2 \/ b3) c= b2 & b5 /\ (b2 \/ b3) c= b3 & b6 /\ (b2 \/ b3) c= b2 /\ b3 & the carrier of b1 = (b4 \/ b5) \/ b6 & b4 is open(b1) & b5 is open(b1) & b6 is closed(b1);

:: TSEP_1:th 64
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2,b3 are_weakly_separated & not b2 c= b3 & not b3 c= b2
   holds ex b4, b5 being non empty Element of bool the carrier of b1 st
      b4 is open(b1) &
       b5 is open(b1) &
       b4 /\ (b2 \/ b3) c= b2 &
       b5 /\ (b2 \/ b3) c= b3 &
       (not b2 \/ b3 c= b4 \/ b5 implies ex b6 being non empty Element of bool the carrier of b1 st
          b6 is closed(b1) & b6 /\ (b2 \/ b3) c= b2 /\ b3 & the carrier of b1 = (b4 \/ b5) \/ b6);

:: TSEP_1:th 65
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 \/ b3 = the carrier of b1
   holds    b2,b3 are_weakly_separated
   iff
      ex b4, b5, b6 being Element of bool the carrier of b1 st
         b2 \/ b3 = (b4 \/ b5) \/ b6 & b4 c= b2 & b5 c= b3 & b6 c= b2 /\ b3 & b4 is open(b1) & b5 is open(b1) & b6 is closed(b1);

:: TSEP_1:th 66
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1
      st b2 \/ b3 = the carrier of b1 & b2,b3 are_weakly_separated & not b2 c= b3 & not b3 c= b2
   holds ex b4, b5 being non empty Element of bool the carrier of b1 st
      b4 is open(b1) &
       b5 is open(b1) &
       b4 c= b2 &
       b5 c= b3 &
       (b2 \/ b3 <> b4 \/ b5 implies ex b6 being non empty Element of bool the carrier of b1 st
          b2 \/ b3 = (b4 \/ b5) \/ b6 & b6 c= b2 /\ b3 & b6 is closed(b1));

:: TSEP_1:th 67
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of bool the carrier of b1 holds
   b2,b3 are_separated
iff
   ex b4, b5 being Element of bool the carrier of b1 st
      b4,b5 are_weakly_separated & b2 c= b4 & b3 c= b5 & b4 /\ b5 misses b2 \/ b3;

:: TSEP_1:prednot 6 => TSEP_1:pred 3
definition
  let a1 be TopStruct;
  let a2, a3 be SubSpace of a1;
  pred A2,A3 are_separated 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 are_separated;
  symmetry;
::  for a1 being TopStruct
::  for a2, a3 being SubSpace of a1
::        st a2,a3 are_separated
::     holds a3,a2 are_separated;
end;

:: TSEP_1:dfs 6
definiens
  let a1 be TopStruct;
  let a2, a3 be SubSpace of a1;
To prove
     a2,a3 are_separated
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 are_separated;

:: TSEP_1:def 8
theorem
for b1 being TopStruct
for b2, b3 being SubSpace of b1 holds
   b2,b3 are_separated
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 are_separated;

:: TSEP_1:prednot 7 => not TSEP_1:pred 3
notation
  let a1 be TopStruct;
  let a2, a3 be SubSpace of a1;
  antonym a2,a3 are_not_separated for a2,a3 are_separated;
end;

:: TSEP_1:th 68
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2,b3 are_separated
   holds b2 misses b3;

:: TSEP_1:th 70
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty closed SubSpace of b1 holds
   b2 misses b3
iff
   b2,b3 are_separated;

:: TSEP_1:th 71
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b1 = b2 union b3 & b2,b3 are_separated
   holds b2 is closed SubSpace of b1;

:: TSEP_1:th 72
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2 union b3 is closed SubSpace of b1 & b2,b3 are_separated
   holds b2 is closed SubSpace of b1;

:: TSEP_1:th 73
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty open SubSpace of b1 holds
   b2 misses b3
iff
   b2,b3 are_separated;

:: TSEP_1:th 74
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b1 = b2 union b3 & b2,b3 are_separated
   holds b2 is open SubSpace of b1;

:: TSEP_1:th 75
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2 union b3 is open SubSpace of b1 & b2,b3 are_separated
   holds b2 is open SubSpace of b1;

:: TSEP_1:th 76
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 & b3,b4 are_separated
   holds b3 meet b2,b4 meet b2 are_separated & b2 meet b3,b2 meet b4 are_separated;

:: TSEP_1:th 77
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
for b4, b5 being SubSpace of b1
      st b4 is SubSpace of b2 & b5 is SubSpace of b3 & b2,b3 are_separated
   holds b4,b5 are_separated;

:: TSEP_1:th 78
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
      st b2 meets b3 & b2,b4 are_separated
   holds b2 meet b3,b4 are_separated;

:: TSEP_1:th 79
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1 holds
   b2,b4 are_separated & b3,b4 are_separated
iff
   b2 union b3,b4 are_separated;

:: TSEP_1:th 80
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
   b2,b3 are_separated
iff
   ex b4, b5 being non empty closed SubSpace of b1 st
      b2 is SubSpace of b4 & b3 is SubSpace of b5 & b4 misses b3 & b5 misses b2;

:: TSEP_1:th 81
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
   b2,b3 are_separated
iff
   ex b4, b5 being non empty closed SubSpace of b1 st
      b2 is SubSpace of b4 & b3 is SubSpace of b5 & (b4 misses b5 or b4 meet b5 misses b2 union b3);

:: TSEP_1:th 82
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
   b2,b3 are_separated
iff
   ex b4, b5 being non empty open SubSpace of b1 st
      b2 is SubSpace of b4 & b3 is SubSpace of b5 & b4 misses b3 & b5 misses b2;

:: TSEP_1:th 83
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
   b2,b3 are_separated
iff
   ex b4, b5 being non empty open SubSpace of b1 st
      b2 is SubSpace of b4 & b3 is SubSpace of b5 & (b4 misses b5 or b4 meet b5 misses b2 union b3);

:: TSEP_1:prednot 8 => TSEP_1:pred 4
definition
  let a1 be TopStruct;
  let a2, a3 be SubSpace of a1;
  pred A2,A3 are_weakly_separated 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 are_weakly_separated;
  symmetry;
::  for a1 being TopStruct
::  for a2, a3 being SubSpace of a1
::        st a2,a3 are_weakly_separated
::     holds a3,a2 are_weakly_separated;
end;

:: TSEP_1:dfs 7
definiens
  let a1 be TopStruct;
  let a2, a3 be SubSpace of a1;
To prove
     a2,a3 are_weakly_separated
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 are_weakly_separated;

:: TSEP_1:def 9
theorem
for b1 being TopStruct
for b2, b3 being SubSpace of b1 holds
   b2,b3 are_weakly_separated
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 are_weakly_separated;

:: TSEP_1:prednot 9 => not TSEP_1:pred 4
notation
  let a1 be TopStruct;
  let a2, a3 be SubSpace of a1;
  antonym a2,a3 are_not_weakly_separated for a2,a3 are_weakly_separated;
end;

:: TSEP_1:th 85
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
   b2 misses b3 & b2,b3 are_weakly_separated
iff
   b2,b3 are_separated;

:: TSEP_1:th 86
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2 is SubSpace of b3
   holds b2,b3 are_weakly_separated;

:: TSEP_1:th 87
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being closed SubSpace of b1 holds
b2,b3 are_weakly_separated;

:: TSEP_1:th 88
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being open SubSpace of b1 holds
b2,b3 are_weakly_separated;

:: TSEP_1:th 89
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
      st b2,b3 are_weakly_separated
   holds b2 union b4,b3 union b4 are_weakly_separated;

:: TSEP_1:th 90
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 & b3,b2 are_weakly_separated
   holds b3 union b4,b2 union b5 are_weakly_separated & b4 union b3,b5 union b2 are_weakly_separated;

:: TSEP_1:th 91
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1
      st b3 meets b4
   holds (b3,b2 are_weakly_separated & b4,b2 are_weakly_separated implies b3 meet b4,b2 are_weakly_separated) &
    (b2,b3 are_weakly_separated & b2,b4 are_weakly_separated implies b2,b3 meet b4 are_weakly_separated);

:: TSEP_1:th 92
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3, b4 being non empty SubSpace of b1 holds
(b2,b4 are_weakly_separated & b3,b4 are_weakly_separated implies b2 union b3,b4 are_weakly_separated) &
 (b4,b2 are_weakly_separated & b4,b3 are_weakly_separated implies b4,b2 union b3 are_weakly_separated);

:: TSEP_1:th 93
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2 meets b3
   holds    b2,b3 are_weakly_separated
   iff
      (b2 is not SubSpace of b3 & b3 is not SubSpace of b2 implies ex b4, b5 being non empty closed SubSpace of b1 st
         b4 meet (b2 union b3) is SubSpace of b2 &
          b5 meet (b2 union b3) is SubSpace of b3 &
          (b2 union b3 is not SubSpace of b4 union b5 implies ex b6 being non empty open SubSpace of b1 st
             TopStruct(#the carrier of b1,the topology of b1#) = (b4 union b5) union b6 &
              b6 meet (b2 union b3) is SubSpace of b2 meet b3));

:: TSEP_1:th 94
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b1 = b2 union b3 & b2 meets b3
   holds    b2,b3 are_weakly_separated
   iff
      (b2 is not SubSpace of b3 & b3 is not SubSpace of b2 implies ex b4, b5 being non empty closed SubSpace of b1 st
         b4 is SubSpace of b2 &
          b5 is SubSpace of b3 &
          (b1 <> b4 union b5 implies ex b6 being non empty open SubSpace of b1 st
             b1 = (b4 union b5) union b6 & b6 is SubSpace of b2 meet b3));

:: TSEP_1:th 95
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b1 = b2 union b3 & b2 misses b3
   holds    b2,b3 are_weakly_separated
   iff
      b2 is closed SubSpace of b1 & b3 is closed SubSpace of b1;

:: TSEP_1:th 96
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b2 meets b3
   holds    b2,b3 are_weakly_separated
   iff
      (b2 is not SubSpace of b3 & b3 is not SubSpace of b2 implies ex b4, b5 being non empty open SubSpace of b1 st
         b4 meet (b2 union b3) is SubSpace of b2 &
          b5 meet (b2 union b3) is SubSpace of b3 &
          (b2 union b3 is not SubSpace of b4 union b5 implies ex b6 being non empty closed SubSpace of b1 st
             TopStruct(#the carrier of b1,the topology of b1#) = (b4 union b5) union b6 &
              b6 meet (b2 union b3) is SubSpace of b2 meet b3));

:: TSEP_1:th 97
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b1 = b2 union b3 & b2 meets b3
   holds    b2,b3 are_weakly_separated
   iff
      (b2 is not SubSpace of b3 & b3 is not SubSpace of b2 implies ex b4, b5 being non empty open SubSpace of b1 st
         b4 is SubSpace of b2 &
          b5 is SubSpace of b3 &
          (b1 <> b4 union b5 implies ex b6 being non empty closed SubSpace of b1 st
             b1 = (b4 union b5) union b6 & b6 is SubSpace of b2 meet b3));

:: TSEP_1:th 98
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1
      st b1 = b2 union b3 & b2 misses b3
   holds    b2,b3 are_weakly_separated
   iff
      b2 is open SubSpace of b1 & b3 is open SubSpace of b1;

:: TSEP_1:th 99
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being non empty SubSpace of b1 holds
   b2,b3 are_separated
iff
   ex b4, b5 being non empty SubSpace of b1 st
      b4,b5 are_weakly_separated & b2 is SubSpace of b4 & b3 is SubSpace of b5 & (b4 misses b5 or b4 meet b5 misses b2 union b3);

:: TSEP_1:th 100
theorem
for b1 being TopStruct holds
   b1 | [#] b1 = TopStruct(#the carrier of b1,the topology of b1#);