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;