Article TOPMETR3, MML version 4.99.1005
:: TOPMETR3:th 1
theorem
for b1 being non empty Element of bool REAL
for b2 being real set
st for b3 being real set
st b3 in b1
holds b3 <= b2
holds upper_bound b1 <= b2;
:: TOPMETR3:th 2
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being Element of bool the carrier of TopSpaceMetr b1
st b2 is convergent(b1) &
(for b4 being Element of NAT holds
b2 . b4 in b3) &
b3 is closed(TopSpaceMetr b1)
holds lim b2 in b3;
:: TOPMETR3:th 3
theorem
for b1, b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being Function-like quasi_total Relation of the carrier of TopSpaceMetr b1,the carrier of TopSpaceMetr b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b3 * b4 is Function-like quasi_total Relation of NAT,the carrier of b2;
:: TOPMETR3:th 4
theorem
for b1, b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being Function-like quasi_total Relation of the carrier of TopSpaceMetr b1,the carrier of TopSpaceMetr b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
for b5 being Function-like quasi_total Relation of NAT,the carrier of b2
st b4 is convergent(b1) & b5 = b3 * b4 & b3 is continuous(TopSpaceMetr b1, TopSpaceMetr b2)
holds b5 is convergent(b2);
:: TOPMETR3:th 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total Relation of NAT,the carrier of RealSpace
st b1 = b2
holds (b1 is convergent implies b2 is convergent(RealSpace)) & (b2 is convergent(RealSpace) implies b1 is convergent) & (b1 is convergent implies lim b1 = lim b2);
:: TOPMETR3:th 7
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
st rng b3 c= [.b1,b2.]
holds b3 is Function-like quasi_total Relation of NAT,the carrier of Closed-Interval-MSpace(b1,b2);
:: TOPMETR3:th 8
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,the carrier of Closed-Interval-MSpace(b1,b2)
st b1 <= b2
holds b3 is Function-like quasi_total Relation of NAT,the carrier of RealSpace;
:: TOPMETR3:th 9
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,the carrier of Closed-Interval-MSpace(b1,b2)
for b4 being Function-like quasi_total Relation of NAT,the carrier of RealSpace
st b4 = b3 & b1 <= b2
holds (b4 is convergent(RealSpace) implies b3 is convergent(Closed-Interval-MSpace(b1,b2))) &
(b3 is convergent(Closed-Interval-MSpace(b1,b2)) implies b4 is convergent(RealSpace)) &
(b4 is convergent(RealSpace) implies lim b4 = lim b3);
:: TOPMETR3:th 10
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of Closed-Interval-MSpace(b1,b2)
st b4 = b3 & b1 <= b2 & b3 is convergent
holds b4 is convergent(Closed-Interval-MSpace(b1,b2)) & lim b3 = lim b4;
:: TOPMETR3:th 11
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of Closed-Interval-MSpace(b1,b2)
st b4 = b3 & b1 <= b2 & b3 is non-decreasing
holds b4 is convergent(Closed-Interval-MSpace(b1,b2));
:: TOPMETR3:th 12
theorem
for b1, b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
for b4 being Function-like quasi_total Relation of NAT,the carrier of Closed-Interval-MSpace(b1,b2)
st b4 = b3 & b1 <= b2 & b3 is non-increasing
holds b4 is convergent(Closed-Interval-MSpace(b1,b2));
:: TOPMETR3:th 15
theorem
for b1 being non empty Element of bool REAL
st b1 is bounded_above
holds ex b2 being Function-like quasi_total Relation of NAT,REAL st
b2 is non-decreasing & b2 is convergent & rng b2 c= b1 & lim b2 = upper_bound b1;
:: TOPMETR3:th 16
theorem
for b1 being non empty Element of bool REAL
st b1 is bounded_below
holds ex b2 being Function-like quasi_total Relation of NAT,REAL st
b2 is non-increasing & b2 is convergent & rng b2 c= b1 & lim b2 = lower_bound b1;
:: TOPMETR3:th 17
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TopSpaceMetr b1
for b3, b4 being Element of bool the carrier of TopSpaceMetr b1
for b5, b6 being Element of REAL
st 0 <= b5 & b6 <= 1 & b5 <= b6 & b2 . b5 in b3 & b2 . b6 in b4 & b3 is closed(TopSpaceMetr b1) & b4 is closed(TopSpaceMetr b1) & b2 is continuous(I[01], TopSpaceMetr b1) & b3 \/ b4 = the carrier of b1
holds ex b7 being Element of REAL st
b5 <= b7 & b7 <= b6 & b2 . b7 in b3 /\ b4;
:: TOPMETR3:th 18
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4, b5 being non empty Element of bool the carrier of TOP-REAL b1
st b4 is_an_arc_of b2,b3 & b5 is_an_arc_of b3,b2 & b5 c= b4
holds b5 = b4;
:: TOPMETR3:th 19
theorem
for b1, b2 being non empty compact Element of bool the carrier of TOP-REAL 2
st b1 is being_simple_closed_curve & b2 is_an_arc_of W-min b1,E-max b1 & b2 c= b1 & b2 <> Upper_Arc b1
holds b2 = Lower_Arc b1;