Article RCOMP_1, MML version 4.99.1005

:: RCOMP_1:sch 1
scheme RCOMP_1:sch 1
ex b1 being Function-like quasi_total Relation of NAT,REAL st
   for b2 being Element of NAT holds
      P1[b2, b1 . b2]
provided
   for b1 being Element of NAT holds
      ex b2 being real set st
         P1[b1, b2];


:: RCOMP_1:th 1
theorem
for b1, b2 being Element of bool REAL
      st for b3 being real set
              st b3 in b1
           holds b3 in b2
   holds b1 c= b2;

:: RCOMP_1:th 3
theorem
for b1, b2 being Element of bool REAL
      st b1 c= b2 & b2 is bounded_below
   holds b1 is bounded_below;

:: RCOMP_1:th 4
theorem
for b1, b2 being Element of bool REAL
      st b1 c= b2 & b2 is bounded_above
   holds b1 is bounded_above;

:: RCOMP_1:th 5
theorem
for b1, b2 being Element of bool REAL
      st b1 c= b2 & b2 is bounded
   holds b1 is bounded;

:: RCOMP_1:funcnot 1 => RCOMP_1:func 1
definition
  let a1, a2 be real set;
  redefine func [.A1,A2.] -> Element of bool REAL equals
    {b1 where b1 is Element of REAL: a1 <= b1 & b1 <= a2};
end;

:: RCOMP_1:def 1
theorem
for b1, b2 being real set holds
[.b1,b2.] = {b3 where b3 is Element of REAL: b1 <= b3 & b3 <= b2};

:: RCOMP_1:funcnot 2 => RCOMP_1:func 2
definition
  let a1, a2 be ext-real set;
  redefine func ].A1,A2.[ -> Element of bool REAL equals
    {b1 where b1 is Element of REAL: a1 < b1 & b1 < a2};
end;

:: RCOMP_1:def 2
theorem
for b1, b2 being ext-real set holds
].b1,b2.[ = {b3 where b3 is Element of REAL: b1 < b3 & b3 < b2};

:: RCOMP_1:th 8
theorem
for b1, b2, b3 being real set holds
   b1 in ].b2 - b3,b2 + b3.[
iff
   abs (b1 - b2) < b3;

:: RCOMP_1:th 9
theorem
for b1, b2, b3 being real set holds
   b1 in [.b2,b3.]
iff
   abs ((b2 + b3) - (2 * b1)) <= b3 - b2;

:: RCOMP_1:th 10
theorem
for b1, b2, b3 being real set holds
   b1 in ].b2,b3.[
iff
   abs ((b2 + b3) - (2 * b1)) < b3 - b2;

:: RCOMP_1:th 11
theorem
for b1, b2 being real set
      st b1 <= b2
   holds [.b1,b2.] = ].b1,b2.[ \/ {b1,b2};

:: RCOMP_1:th 12
theorem
for b1, b2 being real set
      st b1 <= b2
   holds ].b2,b1.[ = {};

:: RCOMP_1:th 13
theorem
for b1, b2 being real set
      st b1 < b2
   holds [.b2,b1.] = {};

:: RCOMP_1:th 14
theorem
for b1 being real set holds
   [.b1,b1.] = {b1};

:: RCOMP_1:th 15
theorem
for b1, b2 being real set holds
(b1 < b2 implies ].b1,b2.[ <> {}) &
 (b1 <= b2 implies b1 in [.b1,b2.] & b2 in [.b1,b2.]) &
 ].b1,b2.[ c= [.b1,b2.];

:: RCOMP_1:th 16
theorem
for b1, b2, b3, b4 being real set
      st b1 in [.b2,b3.] & b4 in [.b2,b3.]
   holds [.b1,b4.] c= [.b2,b3.];

:: RCOMP_1:th 17
theorem
for b1, b2, b3, b4 being real set
      st b1 in ].b2,b3.[ & b4 in ].b2,b3.[
   holds [.b1,b4.] c= ].b2,b3.[;

:: RCOMP_1:th 18
theorem
for b1, b2 being real set
      st b1 <= b2
   holds [.b1,b2.] = [.b1,b2.] \/ [.b2,b1.];

:: RCOMP_1:attrnot 1 => RCOMP_1:attr 1
definition
  let a1 be Element of bool REAL;
  attr a1 is compact means
    for b1 being Function-like quasi_total Relation of NAT,REAL
          st proj2 b1 c= a1
       holds ex b2 being Function-like quasi_total Relation of NAT,REAL st
          b2 is subsequence of b1 & b2 is convergent & lim b2 in a1;
end;

:: RCOMP_1:dfs 3
definiens
  let a1 be Element of bool REAL;
To prove
     a1 is compact
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of NAT,REAL
          st proj2 b1 c= a1
       holds ex b2 being Function-like quasi_total Relation of NAT,REAL st
          b2 is subsequence of b1 & b2 is convergent & lim b2 in a1;

:: RCOMP_1:def 3
theorem
for b1 being Element of bool REAL holds
      b1 is compact
   iff
      for b2 being Function-like quasi_total Relation of NAT,REAL
            st proj2 b2 c= b1
         holds ex b3 being Function-like quasi_total Relation of NAT,REAL st
            b3 is subsequence of b2 & b3 is convergent & lim b3 in b1;

:: RCOMP_1:attrnot 2 => RCOMP_1:attr 2
definition
  let a1 be Element of bool REAL;
  attr a1 is closed means
    for b1 being Function-like quasi_total Relation of NAT,REAL
          st proj2 b1 c= a1 & b1 is convergent
       holds lim b1 in a1;
end;

:: RCOMP_1:dfs 4
definiens
  let a1 be Element of bool REAL;
To prove
     a1 is closed
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of NAT,REAL
          st proj2 b1 c= a1 & b1 is convergent
       holds lim b1 in a1;

:: RCOMP_1:def 4
theorem
for b1 being Element of bool REAL holds
      b1 is closed
   iff
      for b2 being Function-like quasi_total Relation of NAT,REAL
            st proj2 b2 c= b1 & b2 is convergent
         holds lim b2 in b1;

:: RCOMP_1:attrnot 3 => RCOMP_1:attr 3
definition
  let a1 be Element of bool REAL;
  attr a1 is open means
    a1 ` is closed;
end;

:: RCOMP_1:dfs 5
definiens
  let a1 be Element of bool REAL;
To prove
     a1 is open
it is sufficient to prove
  thus a1 ` is closed;

:: RCOMP_1:def 5
theorem
for b1 being Element of bool REAL holds
      b1 is open
   iff
      b1 ` is closed;

:: RCOMP_1:th 22
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
      st proj2 b3 c= [.b1,b2.]
   holds b3 is bounded;

:: RCOMP_1:th 23
theorem
for b1, b2 being real set holds
[.b1,b2.] is closed;

:: RCOMP_1:th 24
theorem
for b1, b2 being real set holds
[.b1,b2.] is compact;

:: RCOMP_1:th 25
theorem
for b1, b2 being real set holds
].b1,b2.[ is open;

:: RCOMP_1:funcreg 1
registration
  let a1, a2 be real set;
  cluster ].a1,a2.[ -> open;
end;

:: RCOMP_1:funcreg 2
registration
  let a1, a2 be real set;
  cluster [.a1,a2.] -> closed;
end;

:: RCOMP_1:th 26
theorem
for b1 being Element of bool REAL
      st b1 is compact
   holds b1 is closed;

:: RCOMP_1:condreg 1
registration
  cluster compact -> closed (Element of bool REAL);
end;

:: RCOMP_1:th 27
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of bool REAL
   st for b3 being real set
           st b3 in b2
        holds ex b4 being real set st
           ex b5 being Element of NAT st
              0 < b4 &
               (for b6 being Element of NAT
                     st b5 < b6
                  holds b4 < abs ((b1 . b6) - b3))
for b3 being Function-like quasi_total Relation of NAT,REAL
      st b3 is subsequence of b1 & b3 is convergent
   holds not lim b3 in b2;

:: RCOMP_1:th 28
theorem
for b1 being Element of bool REAL
      st b1 is compact
   holds b1 is bounded;

:: RCOMP_1:th 29
theorem
for b1 being Element of bool REAL
      st b1 is bounded & b1 is closed
   holds b1 is compact;

:: RCOMP_1:th 30
theorem
for b1 being Element of bool REAL
      st b1 <> {} & b1 is closed & b1 is bounded_above
   holds upper_bound b1 in b1;

:: RCOMP_1:th 31
theorem
for b1 being Element of bool REAL
      st b1 <> {} & b1 is closed & b1 is bounded_below
   holds lower_bound b1 in b1;

:: RCOMP_1:th 32
theorem
for b1 being Element of bool REAL
      st b1 <> {} & b1 is compact
   holds upper_bound b1 in b1 & lower_bound b1 in b1;

:: RCOMP_1:th 33
theorem
for b1 being Element of bool REAL
      st b1 is compact &
         (for b2, b3 being real set
               st b2 in b1 & b3 in b1
            holds [.b2,b3.] c= b1)
   holds ex b2, b3 being real set st
      b1 = [.b2,b3.];

:: RCOMP_1:exreg 1
registration
  cluster complex-membered ext-real-membered real-membered open Element of bool REAL;
end;

:: RCOMP_1:modenot 1 => RCOMP_1:mode 1
definition
  let a1 be real set;
  mode Neighbourhood of A1 -> Element of bool REAL means
    ex b1 being real set st
       0 < b1 & it = ].a1 - b1,a1 + b1.[;
end;

:: RCOMP_1:dfs 6
definiens
  let a1 be real set;
  let a2 be Element of bool REAL;
To prove
     a2 is Neighbourhood of a1
it is sufficient to prove
  thus ex b1 being real set st
       0 < b1 & a2 = ].a1 - b1,a1 + b1.[;

:: RCOMP_1:def 7
theorem
for b1 being real set
for b2 being Element of bool REAL holds
      b2 is Neighbourhood of b1
   iff
      ex b3 being real set st
         0 < b3 & b2 = ].b1 - b3,b1 + b3.[;

:: RCOMP_1:condreg 2
registration
  let a1 be real set;
  cluster -> open (Neighbourhood of a1);
end;

:: RCOMP_1:th 37
theorem
for b1 being real set
for b2 being Neighbourhood of b1 holds
   b1 in b2;

:: RCOMP_1:th 38
theorem
for b1 being real set
for b2, b3 being Neighbourhood of b1 holds
ex b4 being Neighbourhood of b1 st
   b4 c= b2 & b4 c= b3;

:: RCOMP_1:th 39
theorem
for b1 being open Element of bool REAL
for b2 being real set
      st b2 in b1
   holds ex b3 being Neighbourhood of b2 st
      b3 c= b1;

:: RCOMP_1:th 40
theorem
for b1 being open Element of bool REAL
for b2 being real set
      st b2 in b1
   holds ex b3 being real set st
      0 < b3 & ].b2 - b3,b2 + b3.[ c= b1;

:: RCOMP_1:th 41
theorem
for b1 being Element of bool REAL
      st for b2 being real set
              st b2 in b1
           holds ex b3 being Neighbourhood of b2 st
              b3 c= b1
   holds b1 is open;

:: RCOMP_1:th 42
theorem
for b1 being Element of bool REAL holds
      for b2 being real set
            st b2 in b1
         holds ex b3 being Neighbourhood of b2 st
            b3 c= b1
   iff
      b1 is open;

:: RCOMP_1:th 43
theorem
for b1 being Element of bool REAL
      st b1 is open & b1 is bounded_above
   holds not upper_bound b1 in b1;

:: RCOMP_1:th 44
theorem
for b1 being Element of bool REAL
      st b1 is open & b1 is bounded_below
   holds not lower_bound b1 in b1;

:: RCOMP_1:th 45
theorem
for b1 being Element of bool REAL
      st b1 is open &
         b1 is bounded &
         (for b2, b3 being real set
               st b2 in b1 & b3 in b1
            holds [.b2,b3.] c= b1)
   holds ex b2, b3 being real set st
      b1 = ].b2,b3.[;

:: RCOMP_1:th 46
theorem
for b1, b2 being real set holds
].b1,b2.[ misses {b1,b2};

:: RCOMP_1:th 47
theorem
for b1, b2, b3 being real set holds
   b3 in ].b1,b2.[
iff
   b1 < b3 & b3 < b2;

:: RCOMP_1:th 48
theorem
for b1, b2, b3 being real set holds
   b3 in [.b1,b2.]
iff
   b1 <= b3 & b3 <= b2;

:: RCOMP_1:th 49
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds [.b2,b3.] c= [.b1,b4.];

:: RCOMP_1:th 50
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4 & b2 <= b3
   holds [.b1,b3.] \/ [.b2,b4.] = [.b1,b4.];

:: RCOMP_1:th 51
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4 & b2 <= b3
   holds [.b1,b3.] /\ [.b2,b4.] = [.b2,b3.];

:: RCOMP_1:th 52
theorem
for b1, b2, b3, b4 being real set
      st b1 <= b2 & b3 <= b4
   holds ].b2,b3.[ c= ].b1,b4.[;