Article URYSOHN2, MML version 4.99.1005

:: URYSOHN2:th 1
theorem
for b1 being interval Element of bool REAL
      st b1 <> {}
   holds (b1 ^^ <= ^^ b1 or vol b1 = b1 ^^ - ^^ b1) &
    (b1 ^^ = ^^ b1 implies vol b1 = 0.);

:: URYSOHN2:th 2
theorem
for b1 being Element of bool REAL
for b2 being Element of REAL
      st b2 <> 0
   holds b2 " ** (b2 ** b1) = b1;

:: URYSOHN2:th 3
theorem
for b1 being Element of REAL
   st b1 <> 0
for b2 being Element of bool REAL
      st b2 = REAL
   holds b1 ** b2 = b2;

:: URYSOHN2:th 4
theorem
for b1 being Element of bool REAL
      st b1 <> {}
   holds 0 ** b1 = {0};

:: URYSOHN2:th 5
theorem
for b1 being Element of REAL holds
   b1 ** {} = {};

:: URYSOHN2:th 6
theorem
for b1, b2 being Element of ExtREAL
      st b1 <= b2 & (b1 = -infty implies b2 <> -infty) & (b1 = -infty implies not b2 in REAL) & (b1 = -infty implies b2 <> +infty) & (b1 in REAL implies not b2 in REAL) & (b1 in REAL implies b2 <> +infty)
   holds b1 = +infty & b2 = +infty;

:: URYSOHN2:th 7
theorem
for b1 being Element of ExtREAL holds
   [.b1,b1.] is interval Element of bool REAL;

:: URYSOHN2:th 8
theorem
for b1 being interval Element of bool REAL holds
   0 ** b1 is interval Element of bool REAL;

:: URYSOHN2:th 9
theorem
for b1 being interval Element of bool REAL
for b2 being Element of REAL
      st b2 <> 0 & b1 is open_interval
   holds b2 ** b1 is open_interval;

:: URYSOHN2:th 10
theorem
for b1 being interval Element of bool REAL
for b2 being Element of REAL
      st b2 <> 0 & b1 is closed_interval
   holds b2 ** b1 is closed_interval;

:: URYSOHN2:th 11
theorem
for b1 being interval Element of bool REAL
for b2 being Element of REAL
      st 0 < b2 & b1 is right_open_interval
   holds b2 ** b1 is right_open_interval;

:: URYSOHN2:th 12
theorem
for b1 being interval Element of bool REAL
for b2 being Element of REAL
      st b2 < 0 & b1 is right_open_interval
   holds b2 ** b1 is left_open_interval;

:: URYSOHN2:th 13
theorem
for b1 being interval Element of bool REAL
for b2 being Element of REAL
      st 0 < b2 & b1 is left_open_interval
   holds b2 ** b1 is left_open_interval;

:: URYSOHN2:th 14
theorem
for b1 being interval Element of bool REAL
for b2 being Element of REAL
      st b2 < 0 & b1 is left_open_interval
   holds b2 ** b1 is right_open_interval;

:: URYSOHN2:th 15
theorem
for b1 being interval Element of bool REAL
   st b1 <> {}
for b2 being Element of REAL
   st 0 < b2
for b3 being interval Element of bool REAL
      st b3 = b2 ** b1 & b1 = [.^^ b1,b1 ^^.]
   holds b3 = [.^^ b3,b3 ^^.] &
    (for b4, b5 being Element of REAL
          st b4 = ^^ b1 & b5 = b1 ^^
       holds ^^ b3 = b2 * b4 & b3 ^^ = b2 * b5);

:: URYSOHN2:th 16
theorem
for b1 being interval Element of bool REAL
   st b1 <> {}
for b2 being Element of REAL
   st 0 < b2
for b3 being interval Element of bool REAL
      st b3 = b2 ** b1 & b1 = ].^^ b1,b1 ^^.]
   holds b3 = ].^^ b3,b3 ^^.] &
    (for b4, b5 being Element of REAL
          st b4 = ^^ b1 & b5 = b1 ^^
       holds ^^ b3 = b2 * b4 & b3 ^^ = b2 * b5);

:: URYSOHN2:th 17
theorem
for b1 being interval Element of bool REAL
   st b1 <> {}
for b2 being Element of REAL
   st 0 < b2
for b3 being interval Element of bool REAL
      st b3 = b2 ** b1 & b1 = ].^^ b1,b1 ^^.[
   holds b3 = ].^^ b3,b3 ^^.[ &
    (for b4, b5 being Element of REAL
          st b4 = ^^ b1 & b5 = b1 ^^
       holds ^^ b3 = b2 * b4 & b3 ^^ = b2 * b5);

:: URYSOHN2:th 18
theorem
for b1 being interval Element of bool REAL
   st b1 <> {}
for b2 being Element of REAL
   st 0 < b2
for b3 being interval Element of bool REAL
      st b3 = b2 ** b1 & b1 = [.^^ b1,b1 ^^.[
   holds b3 = [.^^ b3,b3 ^^.[ &
    (for b4, b5 being Element of REAL
          st b4 = ^^ b1 & b5 = b1 ^^
       holds ^^ b3 = b2 * b4 & b3 ^^ = b2 * b5);

:: URYSOHN2:th 19
theorem
for b1 being interval Element of bool REAL
for b2 being Element of REAL holds
   b2 ** b1 is interval Element of bool REAL;

:: URYSOHN2:funcreg 1
registration
  let a1 be interval Element of bool REAL;
  let a2 be Element of REAL;
  cluster a2 ** a1 -> interval;
end;

:: URYSOHN2:th 20
theorem
for b1 being interval Element of bool REAL
for b2 being Element of REAL
   st 0 <= b2
for b3 being Element of REAL
      st b3 = vol b1
   holds b2 * b3 = vol (b2 ** b1);

:: URYSOHN2:th 23
theorem
for b1 being Element of REAL
      st 0 < b1
   holds ex b2 being Element of NAT st
      1 < (2 |^ b2) * b1;

:: URYSOHN2:th 24
theorem
for b1, b2 being Element of REAL
      st 0 <= b1 & 1 < b2 - b1
   holds ex b3 being Element of NAT st
      b1 < b3 & b3 < b2;

:: URYSOHN2:th 27
theorem
for b1 being Element of NAT holds
   dyadic b1 c= DYADIC;

:: URYSOHN2:th 28
theorem
for b1, b2 being Element of REAL
      st b1 < b2 & 0 <= b1 & b2 <= 1
   holds ex b3 being Element of REAL st
      b3 in DYADIC & b1 < b3 & b3 < b2;

:: URYSOHN2:th 29
theorem
for b1, b2 being Element of REAL
      st b1 < b2
   holds ex b3 being Element of REAL st
      b3 in DOM & b1 < b3 & b3 < b2;

:: URYSOHN2:th 30
theorem
for b1 being non empty Element of bool ExtREAL
for b2, b3 being Element of ExtREAL
      st b1 c= [.b2,b3.]
   holds b2 <= inf b1 & sup b1 <= b3;

:: URYSOHN2:th 31
theorem
0 in DYADIC & 1 in DYADIC;

:: URYSOHN2:th 32
theorem
for b1, b2 being Element of ExtREAL
      st b1 = 0 & b2 = 1
   holds DYADIC c= [.b1,b2.];

:: URYSOHN2:th 33
theorem
for b1, b2 being Element of NAT
      st b1 <= b2
   holds dyadic b1 c= dyadic b2;

:: URYSOHN2:th 34
theorem
for b1, b2, b3, b4 being Element of REAL
      st b1 < b3 & b3 < b2 & b1 < b4 & b4 < b2
   holds abs (b4 - b3) < b2 - b1;

:: URYSOHN2:th 35
theorem
for b1 being Element of REAL
   st 0 < b1
for b2 being Element of REAL
      st 0 < b2 & b2 <= 1
   holds ex b3, b4 being Element of REAL st
      b3 in DYADIC \/ right_open_halfline 1 & b4 in DYADIC \/ right_open_halfline 1 & 0 < b3 & b3 < b2 & b2 < b4 & b4 - b3 < b1;