Article JORDAN16, MML version 4.99.1005
:: JORDAN16:th 1
theorem
for b1 being non empty finite Element of bool REAL
for b2 being Element of REAL
st for b3 being Element of REAL
st b3 in b1
holds b3 < b2
holds max b1 < b2;
:: JORDAN16:exreg 1
registration
let a1 be Element of NAT;
cluster trivial Element of bool the carrier of TOP-REAL a1;
end;
:: JORDAN16:th 2
theorem
for b1, b2, b3, b4 being set
st b1 in b4 & b2 in b4 & b3 in b4
holds {b1,b2,b3} c= b4;
:: JORDAN16:th 4
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2 holds
Lower_Arc b1 <> Upper_Arc b1;
:: JORDAN16:th 5
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2 holds
Segment(b1,b2,b3,b4,b5) c= b1;
:: JORDAN16:th 7
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & b4 in b1
holds b4 in L_Segment(b1,b2,b3,b4);
:: JORDAN16:th 8
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & b4 in b1
holds b4 in R_Segment(b1,b2,b3,b4);
:: JORDAN16:th 9
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3
holds b4 in Segment(b1,b2,b3,b4,b5) & b5 in Segment(b1,b2,b3,b4,b5);
:: JORDAN16:th 10
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2 holds
Segment(b2,b3,b1) c= b1;
:: JORDAN16:th 11
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
st b2 in b1 & b3 in b1 & not LE b2,b3,b1
holds LE b3,b2,b1;
:: JORDAN16:th 12
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b2
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b3
st b4 = b5 & b4 is continuous(b1, b2)
holds b5 is continuous(b1, b3);
:: JORDAN16:th 13
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being non empty SubSpace of b1
for b4 being non empty SubSpace of b2
for b5 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b5 is being_homeomorphism(b1, b2)
for b6 being Function-like quasi_total Relation of the carrier of b3,the carrier of b4
st b6 = b5 | b3 & b6 is onto(the carrier of b3, the carrier of b4)
holds b6 is being_homeomorphism(b3, b4);
:: JORDAN16:th 14
theorem
for b1, b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4, b5 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b4,b5 & b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b4,b5 & b2 /\ b3 = {b4,b5} & b1 c= b2 \/ b3 & b1 <> b2
holds b1 = b3;
:: JORDAN16:th 15
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4, b5 being Element of the carrier of TOP-REAL 2
st b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b4,b5 & b2 c= b1 & b3 c= b1 & b2 <> b3
holds b2 \/ b3 = b1 & b2 /\ b3 = {b4,b5};
:: JORDAN16:th 16
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
for b3, b4, b5, b6 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b3,b4 & b1 /\ b2 = {b5,b6}
holds b1 <> b2;
:: JORDAN16:th 17
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4, b5 being Element of the carrier of TOP-REAL 2
st b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b4,b5 & b2 c= b1 & b3 c= b1 & b2 /\ b3 = {b4,b5}
holds b2 \/ b3 = b1;
:: JORDAN16:th 18
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of bool the carrier of TOP-REAL 2
for b4, b5 being Element of the carrier of TOP-REAL 2
st b2 c= b1 & b3 c= b1 & b2 <> b3 & b2 is_an_arc_of b4,b5 & b3 is_an_arc_of b4,b5
for b6 being Element of bool the carrier of TOP-REAL 2
st b6 is_an_arc_of b4,b5 & b6 c= b1 & b6 <> b2
holds b6 = b3;
:: JORDAN16:th 19
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being non empty Element of bool the carrier of TOP-REAL 2
st b2 is_an_arc_of W-min b1,E-max b1 & b2 c= b1 & b2 <> Lower_Arc b1
holds b2 = Upper_Arc b1;
:: JORDAN16:th 20
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3
holds ex b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1 st
ex b7, b8 being Element of REAL st
b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b6 . 0 = b2 & b6 . 1 = b3 & b6 . b7 = b4 & b6 . b8 = b5 & 0 <= b7 & b7 <= b8 & b8 <= 1;
:: JORDAN16:th 21
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 & b4 <> b5
holds ex b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL 2) | b1 st
ex b7, b8 being Element of REAL st
b6 is being_homeomorphism(I[01], (TOP-REAL 2) | b1) & b6 . 0 = b2 & b6 . 1 = b3 & b6 . b7 = b4 & b6 . b8 = b5 & 0 <= b7 & b7 < b8 & b8 <= 1;
:: JORDAN16:th 22
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3
holds Segment(b1,b2,b3,b4,b5) is not empty;
:: JORDAN16:th 23
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in b1
holds b2 in Segment(b2,W-min b1,b1) & W-min b1 in Segment(b2,W-min b1,b1);
:: JORDAN16:attrnot 1 => JORDAN16:attr 1
definition
let a1 be Function-like Relation of REAL,REAL;
attr a1 is continuous means
a1 is_continuous_on dom a1;
end;
:: JORDAN16:dfs 1
definiens
let a1 be Function-like Relation of REAL,REAL;
To prove
a1 is continuous
it is sufficient to prove
thus a1 is_continuous_on dom a1;
:: JORDAN16:def 1
theorem
for b1 being Function-like Relation of REAL,REAL holds
b1 is continuous
iff
b1 is_continuous_on dom b1;
:: JORDAN16:attrnot 2 => JORDAN16:attr 1
definition
let a1 be Function-like Relation of REAL,REAL;
attr a1 is continuous means
a1 is_continuous_on REAL;
end;
:: JORDAN16:dfs 2
definiens
let a1 be Function-like quasi_total Relation of REAL,REAL;
To prove
a1 is continuous
it is sufficient to prove
thus a1 is_continuous_on REAL;
:: JORDAN16:def 2
theorem
for b1 being Function-like quasi_total Relation of REAL,REAL holds
b1 is continuous
iff
b1 is_continuous_on REAL;
:: JORDAN16:funcnot 1 => JORDAN16:func 1
definition
let a1, a2 be real set;
func AffineMap(A1,A2) -> Function-like quasi_total Relation of REAL,REAL means
for b1 being real set holds
it . b1 = (a1 * b1) + a2;
end;
:: JORDAN16:def 3
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of REAL,REAL holds
b3 = AffineMap(b1,b2)
iff
for b4 being real set holds
b3 . b4 = (b1 * b4) + b2;
:: JORDAN16:funcreg 1
registration
let a1, a2 be real set;
cluster AffineMap(a1,a2) -> Function-like quasi_total continuous;
end;
:: JORDAN16:exreg 2
registration
cluster Relation-like Function-like non empty quasi_total total complex-valued ext-real-valued real-valued continuous Relation of REAL,REAL;
end;
:: JORDAN16:th 24
theorem
for b1, b2 being Function-like continuous Relation of REAL,REAL holds
b2 * b1 is Function-like continuous Relation of REAL,REAL;
:: JORDAN16:th 25
theorem
for b1, b2 being real set holds
(AffineMap(b1,b2)) . 0 = b2;
:: JORDAN16:th 26
theorem
for b1, b2 being real set holds
(AffineMap(b1,b2)) . 1 = b1 + b2;
:: JORDAN16:th 27
theorem
for b1, b2 being real set
st b1 <> 0
holds AffineMap(b1,b2) is one-to-one;
:: JORDAN16:th 28
theorem
for b1, b2, b3, b4 being real set
st 0 < b1 & b3 < b4
holds (AffineMap(b1,b2)) . b3 < (AffineMap(b1,b2)) . b4;
:: JORDAN16:th 29
theorem
for b1, b2, b3, b4 being real set
st b1 < 0 & b3 < b4
holds (AffineMap(b1,b2)) . b4 < (AffineMap(b1,b2)) . b3;
:: JORDAN16:th 30
theorem
for b1, b2, b3, b4 being real set
st 0 <= b1 & b3 <= b4
holds (AffineMap(b1,b2)) . b3 <= (AffineMap(b1,b2)) . b4;
:: JORDAN16:th 31
theorem
for b1, b2, b3, b4 being real set
st b1 <= 0 & b3 <= b4
holds (AffineMap(b1,b2)) . b4 <= (AffineMap(b1,b2)) . b3;
:: JORDAN16:th 32
theorem
for b1, b2 being real set
st b1 <> 0
holds rng AffineMap(b1,b2) = REAL;
:: JORDAN16:th 33
theorem
for b1, b2 being real set
st b1 <> 0
holds (AffineMap(b1,b2)) " = AffineMap(b1 ",- (b2 / b1));
:: JORDAN16:th 34
theorem
for b1, b2 being real set
st 0 < b1
holds (AffineMap(b1,b2)) .: [.0,1.] = [.b2,b1 + b2.];
:: JORDAN16:th 35
theorem
for b1 being Function-like quasi_total Relation of the carrier of R^1,the carrier of R^1
for b2, b3 being Element of REAL
st b2 <> 0 & b1 = AffineMap(b2,b3)
holds b1 is being_homeomorphism(R^1, R^1);
:: JORDAN16:th 36
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & LE b4,b5,b1,b2,b3 & b4 <> b5
holds Segment(b1,b2,b3,b4,b5) is_an_arc_of b4,b5;
:: JORDAN16:th 37
theorem
for b1 being being_simple_closed_curve Element of bool the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of bool the carrier of TOP-REAL 2
st b4 c= b1 & b4 is_an_arc_of b2,b3 & W-min b1 in b4 & E-max b1 in b4 & not Upper_Arc b1 c= b4
holds Lower_Arc b1 c= b4;
:: JORDAN16:th 38
theorem
for b1 being Element of bool the carrier of TOP-REAL 2
for b2, b3, b4, b5 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & b4 in b1 & b5 in b1 & b4 <> b2 & b4 <> b3 & b5 <> b2 & b5 <> b3 & b4 <> b5
holds ex b6 being non empty Element of bool the carrier of TOP-REAL 2 st
b6 is_an_arc_of b4,b5 & b6 c= b1 & b6 misses {b2,b3};
:: JORDAN16:th 39
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b1 is_an_arc_of b2,b3 & b4 in b1 & b2 <> b4
holds Segment(b1,b2,b3,b2,b4) is_an_arc_of b2,b4;