Article JCT_MISC, MML version 4.99.1005

:: JCT_MISC:sch 1
scheme JCT_MISC:sch 1
{F1 -> non empty set,
  F2 -> set}:
{F2(b1) where b1 is Element of F1(): TRUE} is not empty


:: JCT_MISC:th 5
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of bool b2
      st (b2 = {} implies b1 = {})
   holds (b3 " b4) ` = b3 " (b4 `);

:: JCT_MISC:th 6
theorem
for b1 being 1-sorted
for b2 being non empty set
for b3 being Function-like quasi_total Relation of the carrier of b1,b2
for b4 being Element of bool b2 holds
   (b3 " b4) ` = b3 " (b4 `);

:: JCT_MISC:th 7
theorem
for b1, b2 being Element of NAT
      st b1 <= b2
   holds b2 -' (b2 -' b1) = b1;

:: JCT_MISC:th 9
theorem
for b1, b2, b3, b4 being real set
      st b1 in [.b3,b4.] & b2 in [.b3,b4.]
   holds (b1 + b2) / 2 in [.b3,b4.];

:: JCT_MISC:th 11
theorem
for b1, b2, b3, b4 being real set holds
abs ((abs (b1 - b2)) - abs (b3 - b4)) <= (abs (b1 - b3)) + abs (b2 - b4);

:: JCT_MISC:th 12
theorem
for b1, b2, b3 being real set
      st b1 in ].b2,b3.[
   holds abs b1 < max(abs b2,abs b3);

:: JCT_MISC:sch 2
scheme JCT_MISC:sch 2
{F1 -> non empty set,
  F2 -> non empty set,
  F3 -> non empty set}:
ex b1 being Function-like quasi_total Relation of F1(),F2() st
   ex b2 being Function-like quasi_total Relation of F1(),F3() st
      for b3 being Element of F1() holds
         P1[b3, b1 . b3, b2 . b3]
provided
   for b1 being Element of F1() holds
      ex b2 being Element of F2() st
         ex b3 being Element of F3() st
            P1[b1, b2, b3];


:: JCT_MISC:th 13
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of bool the carrier of [:b1,b2:]
      st for b4 being Element of the carrier of [:b1,b2:]
              st b4 in b3
           holds ex b5 being Element of bool the carrier of b1 st
              ex b6 being Element of bool the carrier of b2 st
                 b5 is open(b1) & b6 is open(b2) & b4 in [:b5,b6:] & [:b5,b6:] c= b3
   holds b3 is open([:b1,b2:]);

:: JCT_MISC:th 14
theorem
for b1, b2 being compact Element of bool REAL holds
b1 /\ b2 is compact;

:: JCT_MISC:attrnot 1 => JCT_MISC:attr 1
definition
  let a1 be Element of bool REAL;
  attr a1 is connected means
    for b1, b2 being real set
          st b1 in a1 & b2 in a1
       holds [.b1,b2.] c= a1;
end;

:: JCT_MISC:dfs 1
definiens
  let a1 be Element of bool REAL;
To prove
     a1 is connected
it is sufficient to prove
  thus for b1, b2 being real set
          st b1 in a1 & b2 in a1
       holds [.b1,b2.] c= a1;

:: JCT_MISC:def 1
theorem
for b1 being Element of bool REAL holds
      b1 is connected
   iff
      for b2, b3 being real set
            st b2 in b1 & b3 in b1
         holds [.b2,b3.] c= b1;

:: JCT_MISC:th 15
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total continuous Relation of the carrier of b1,REAL
for b3 being Element of bool the carrier of b1
      st b3 is connected(b1)
   holds b2 .: b3 is connected;

:: JCT_MISC:funcnot 1 => JCT_MISC:func 1
definition
  let a1, a2 be Element of bool REAL;
  func dist(A1,A2) -> real set means
    ex b1 being Element of bool REAL st
       b1 = {abs (b2 - b3) where b2 is Element of REAL, b3 is Element of REAL: b2 in a1 & b3 in a2} &
        it = lower_bound b1;
  commutativity;
::  for a1, a2 being Element of bool REAL holds
::  dist(a1,a2) = dist(a2,a1);
end;

:: JCT_MISC:def 2
theorem
for b1, b2 being Element of bool REAL
for b3 being real set holds
      b3 = dist(b1,b2)
   iff
      ex b4 being Element of bool REAL st
         b4 = {abs (b5 - b6) where b5 is Element of REAL, b6 is Element of REAL: b5 in b1 & b6 in b2} &
          b3 = lower_bound b4;

:: JCT_MISC:th 16
theorem
for b1, b2 being Element of bool REAL
for b3, b4 being real set
      st b3 in b1 & b4 in b2
   holds dist(b1,b2) <= abs (b3 - b4);

:: JCT_MISC:th 17
theorem
for b1, b2 being Element of bool REAL
for b3, b4 being non empty Element of bool REAL
      st b3 c= b1 & b4 c= b2
   holds dist(b1,b2) <= dist(b3,b4);

:: JCT_MISC:th 18
theorem
for b1, b2 being non empty compact Element of bool REAL holds
ex b3, b4 being real set st
   b3 in b1 & b4 in b2 & dist(b1,b2) = abs (b3 - b4);

:: JCT_MISC:th 19
theorem
for b1, b2 being non empty compact Element of bool REAL holds
0 <= dist(b1,b2);

:: JCT_MISC:th 20
theorem
for b1, b2 being non empty compact Element of bool REAL
      st b1 misses b2
   holds 0 < dist(b1,b2);

:: JCT_MISC:th 21
theorem
for b1, b2 being real set
for b3, b4 being compact Element of bool REAL
   st b3 misses b4 & b3 c= [.b1,b2.] & b4 c= [.b1,b2.]
for b5 being Function-like quasi_total Relation of NAT,bool REAL
      st for b6 being Element of NAT holds
           b5 . b6 is connected & b5 . b6 meets b3 & b5 . b6 meets b4
   holds ex b6 being real set st
      b6 in [.b1,b2.] &
       not b6 in b3 \/ b4 &
       (for b7 being Element of NAT holds
          ex b8 being Element of NAT st
             b7 <= b8 & b6 in b5 . b8);

:: JCT_MISC:th 22
theorem
for b1 being closed Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds proj2 .: b1 is closed;

:: JCT_MISC:th 23
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
      st b1 is Bounded(2)
   holds proj2 .: b1 is bounded;

:: JCT_MISC:th 24
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2 holds
   proj2 .: b1 is compact;