Article JORDAN5A, MML version 4.99.1005
:: JORDAN5A:th 1
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of bool the carrier of TOP-REAL b1
st b4 is_an_arc_of b2,b3
holds b4 is compact(TOP-REAL b1);
:: JORDAN5A:th 3
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4, b5 being real set
st ((1 - b4) * b2) + (b4 * b3) = ((1 - b5) * b2) + (b5 * b3) &
b4 <> b5
holds b2 = b3;
:: JORDAN5A:th 4
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
st b2 <> b3
holds ex b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL b1) | LSeg(b2,b3) st
(for b5 being Element of REAL
st b5 in [.0,1.]
holds b4 . b5 = ((1 - b5) * b2) + (b5 * b3)) &
b4 is being_homeomorphism(I[01], (TOP-REAL b1) | LSeg(b2,b3)) &
b4 . 0 = b2 &
b4 . 1 = b3;
:: JORDAN5A:funcreg 1
registration
let a1 be Element of NAT;
cluster TOP-REAL a1 -> strict TopSpace-like arcwise_connected;
end;
:: JORDAN5A:exreg 1
registration
let a1 be Element of NAT;
cluster non empty strict TopSpace-like compact SubSpace of TOP-REAL a1;
end;
:: JORDAN5A:th 5
theorem
for b1, b2 being Element of the carrier of TOP-REAL 2
for b3 being Path of b1,b2
for b4 being non empty compact SubSpace of TOP-REAL 2
for b5 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b4
st b3 is one-to-one & b5 = b3 & [#] b4 = rng b3
holds b5 is being_homeomorphism(I[01], b4);
:: JORDAN5A:th 6
theorem
for b1 being Element of bool REAL holds
b1 in Family_open_set RealSpace
iff
b1 is open;
:: JORDAN5A:th 7
theorem
for b1 being Function-like quasi_total Relation of the carrier of R^1,the carrier of R^1
for b2 being Element of the carrier of R^1
for b3 being Function-like Relation of REAL,REAL
for b4 being Element of REAL
st b1 is_continuous_at b2 & b1 = b3 & b2 = b4
holds b3 is_continuous_in b4;
:: JORDAN5A:th 8
theorem
for b1 being Function-like quasi_total continuous Relation of the carrier of R^1,the carrier of R^1
for b2 being Function-like Relation of REAL,REAL
st b1 = b2
holds b2 is_continuous_on REAL;
:: JORDAN5A:th 9
theorem
for b1 being Function-like one-to-one quasi_total continuous Relation of the carrier of R^1,the carrier of R^1
st ex b2, b3 being Element of the carrier of I[01] st
ex b4, b5, b6, b7 being Element of REAL st
b2 = b4 & b3 = b5 & b4 < b5 & b6 = b1 . b2 & b7 = b1 . b3 & b7 <= b6
for b2, b3 being Element of the carrier of I[01]
for b4, b5, b6, b7 being Element of REAL
st b2 = b4 & b3 = b5 & b4 < b5 & b6 = b1 . b2 & b7 = b1 . b3
holds b7 < b6;
:: JORDAN5A:th 10
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Element of the carrier of Closed-Interval-MSpace(b3,b4)
st b3 <= b4 &
b5 = b1 &
0 < b2 &
].b1 - b2,b1 + b2.[ c= [.b3,b4.]
holds ].b1 - b2,b1 + b2.[ = Ball(b5,b2);
:: JORDAN5A:th 11
theorem
for b1, b2 being Element of REAL
for b3 being Element of bool REAL
st b1 < b2 & not b1 in b3 & not b2 in b3 & b3 in Family_open_set Closed-Interval-MSpace(b1,b2)
holds b3 is open;
:: JORDAN5A:th 12
theorem
for b1 being open Element of bool REAL
for b2, b3 being Element of REAL
st b1 c= [.b2,b3.]
holds not b2 in b1 & not b3 in b1;
:: JORDAN5A:th 13
theorem
for b1, b2 being Element of REAL
for b3 being Element of bool REAL
for b4 being Element of bool the carrier of Closed-Interval-MSpace(b1,b2)
st b1 <= b2 & b4 = b3 & b3 is open
holds b4 in Family_open_set Closed-Interval-MSpace(b1,b2);
:: JORDAN5A:th 14
theorem
for b1, b2, b3, b4, b5 being Element of REAL
for b6 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
for b7 being Element of the carrier of Closed-Interval-TSpace(b1,b2)
for b8 being Function-like Relation of REAL,REAL
st b1 < b2 & b3 < b4 & b6 is_continuous_at b7 & b6 . b1 = b3 & b6 . b2 = b4 & b6 is one-to-one & b6 = b8 & b7 = b5
holds b8 | [.b1,b2.] is_continuous_in b5;
:: JORDAN5A:th 15
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
for b6 being Function-like Relation of REAL,REAL
st b5 is continuous(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b5 is one-to-one & b1 < b2 & b3 < b4 & b5 = b6 & b5 . b1 = b3 & b5 . b2 = b4
holds b6 is_continuous_on [.b1,b2.];
:: JORDAN5A:th 16
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
st b1 < b2 & b3 < b4 & b5 is continuous(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b5 is one-to-one & b5 . b1 = b3 & b5 . b2 = b4
for b6, b7 being Element of the carrier of Closed-Interval-TSpace(b1,b2)
for b8, b9, b10, b11 being Element of REAL
st b6 = b8 & b7 = b9 & b8 < b9 & b10 = b5 . b6 & b11 = b5 . b7
holds b10 < b11;
:: JORDAN5A:th 17
theorem
for b1 being Function-like one-to-one quasi_total continuous Relation of the carrier of I[01],the carrier of I[01]
st b1 . 0 = 0 & b1 . 1 = 1
for b2, b3 being Element of the carrier of I[01]
for b4, b5, b6, b7 being Element of REAL
st b2 = b4 & b3 = b5 & b4 < b5 & b6 = b1 . b2 & b7 = b1 . b3
holds b6 < b7;
:: JORDAN5A:th 18
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
for b6 being non empty Element of bool the carrier of Closed-Interval-TSpace(b1,b2)
for b7, b8 being Element of bool the carrier of R^1
st b1 < b2 & b3 < b4 & b7 = b6 & b5 is continuous(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b5 is one-to-one & b7 is compact(R^1) & b5 . b1 = b3 & b5 . b2 = b4 & b5 .: b6 = b8
holds b5 . lower_bound [#] b7 = lower_bound [#] b8;
:: JORDAN5A:th 19
theorem
for b1, b2, b3, b4 being Element of REAL
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
for b6, b7 being non empty Element of bool the carrier of Closed-Interval-TSpace(b1,b2)
for b8, b9 being Element of bool the carrier of R^1
st b1 < b2 & b3 < b4 & b8 = b6 & b9 = b7 & b5 is continuous(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b5 is one-to-one & b8 is compact(R^1) & b5 . b1 = b3 & b5 . b2 = b4 & b5 .: b6 = b7
holds b5 . upper_bound [#] b8 = upper_bound [#] b9;
:: JORDAN5A:th 20
theorem
for b1, b2 being real set
st b1 <= b2
holds lower_bound [.b1,b2.] = b1 & upper_bound [.b1,b2.] = b2;
:: JORDAN5A:th 21
theorem
for b1, b2, b3, b4, b5, b6, b7, b8 being Element of REAL
for b9 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b3,b4)
st b1 < b2 & b3 < b4 & b5 < b6 & b1 <= b5 & b6 <= b2 & b9 is being_homeomorphism(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(b3,b4)) & b9 . b1 = b3 & b9 . b2 = b4 & b7 = b9 . b5 & b8 = b9 . b6
holds b9 .: [.b5,b6.] = [.b7,b8.];
:: JORDAN5A:th 22
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
holds ex b5 being Element of the carrier of TOP-REAL 2 st
b5 in b1 /\ b2 &
(ex b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1 st
ex b7 being Element of REAL st
b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) &
b6 . 0 = b3 &
b6 . 1 = b4 &
b6 . b7 = b5 &
0 <= b7 &
b7 <= 1 &
(for b8 being Element of REAL
st 0 <= b8 & b8 < b7
holds not b6 . b8 in b2));
:: JORDAN5A:th 23
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st b1 meets b2 & b1 /\ b2 is closed(TOP-REAL 2) & b1 is_an_arc_of b3,b4
holds ex b5 being Element of the carrier of TOP-REAL 2 st
b5 in b1 /\ b2 &
(ex b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1 st
ex b7 being Element of REAL st
b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) &
b6 . 0 = b3 &
b6 . 1 = b4 &
b6 . b7 = b5 &
0 <= b7 &
b7 <= 1 &
(for b8 being Element of REAL
st b8 <= 1 & b7 < b8
holds not b6 . b8 in b2));