Article SEQM_3, MML version 4.99.1005
:: SEQM_3:attrnot 1 => SEQM_3:attr 1
definition
let a1 be Function-like Relation of NAT,REAL;
attr a1 is increasing means
for b1, b2 being Element of NAT
st b1 in proj1 a1 & b2 in proj1 a1 & b1 < b2
holds a1 . b1 < a1 . b2;
end;
:: SEQM_3:dfs 1
definiens
let a1 be Function-like Relation of NAT,REAL;
To prove
a1 is increasing
it is sufficient to prove
thus for b1, b2 being Element of NAT
st b1 in proj1 a1 & b2 in proj1 a1 & b1 < b2
holds a1 . b1 < a1 . b2;
:: SEQM_3:def 1
theorem
for b1 being Function-like Relation of NAT,REAL holds
b1 is increasing
iff
for b2, b3 being Element of NAT
st b2 in proj1 b1 & b3 in proj1 b1 & b2 < b3
holds b1 . b2 < b1 . b3;
:: SEQM_3:attrnot 2 => SEQM_3:attr 2
definition
let a1 be Function-like Relation of NAT,REAL;
attr a1 is decreasing means
for b1, b2 being Element of NAT
st b1 in proj1 a1 & b2 in proj1 a1 & b1 < b2
holds a1 . b2 < a1 . b1;
end;
:: SEQM_3:dfs 2
definiens
let a1 be Function-like Relation of NAT,REAL;
To prove
a1 is decreasing
it is sufficient to prove
thus for b1, b2 being Element of NAT
st b1 in proj1 a1 & b2 in proj1 a1 & b1 < b2
holds a1 . b2 < a1 . b1;
:: SEQM_3:def 2
theorem
for b1 being Function-like Relation of NAT,REAL holds
b1 is decreasing
iff
for b2, b3 being Element of NAT
st b2 in proj1 b1 & b3 in proj1 b1 & b2 < b3
holds b1 . b3 < b1 . b2;
:: SEQM_3:attrnot 3 => SEQM_3:attr 3
definition
let a1 be Function-like Relation of NAT,REAL;
attr a1 is non-decreasing means
for b1, b2 being Element of NAT
st b1 in proj1 a1 & b2 in proj1 a1 & b1 <= b2
holds a1 . b1 <= a1 . b2;
end;
:: SEQM_3:dfs 3
definiens
let a1 be Function-like Relation of NAT,REAL;
To prove
a1 is non-decreasing
it is sufficient to prove
thus for b1, b2 being Element of NAT
st b1 in proj1 a1 & b2 in proj1 a1 & b1 <= b2
holds a1 . b1 <= a1 . b2;
:: SEQM_3:def 3
theorem
for b1 being Function-like Relation of NAT,REAL holds
b1 is non-decreasing
iff
for b2, b3 being Element of NAT
st b2 in proj1 b1 & b3 in proj1 b1 & b2 <= b3
holds b1 . b2 <= b1 . b3;
:: SEQM_3:attrnot 4 => SEQM_3:attr 4
definition
let a1 be Function-like Relation of NAT,REAL;
attr a1 is non-increasing means
for b1, b2 being Element of NAT
st b1 in proj1 a1 & b2 in proj1 a1 & b1 <= b2
holds a1 . b2 <= a1 . b1;
end;
:: SEQM_3:dfs 4
definiens
let a1 be Function-like Relation of NAT,REAL;
To prove
a1 is non-increasing
it is sufficient to prove
thus for b1, b2 being Element of NAT
st b1 in proj1 a1 & b2 in proj1 a1 & b1 <= b2
holds a1 . b2 <= a1 . b1;
:: SEQM_3:def 4
theorem
for b1 being Function-like Relation of NAT,REAL holds
b1 is non-increasing
iff
for b2, b3 being Element of NAT
st b2 in proj1 b1 & b3 in proj1 b1 & b2 <= b3
holds b1 . b3 <= b1 . b2;
:: SEQM_3:attrnot 5 => FUNCT_1:attr 3
definition
let a1 be Relation-like Function-like set;
attr a1 is constant means
ex b1 being real set st
for b2 being Element of NAT holds
a1 . b2 = b1;
end;
:: SEQM_3:dfs 5
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is constant
it is sufficient to prove
thus ex b1 being real set st
for b2 being Element of NAT holds
a1 . b2 = b1;
:: SEQM_3:def 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is constant
iff
ex b2 being real set st
for b3 being Element of NAT holds
b1 . b3 = b2;
:: SEQM_3:attrnot 6 => SEQM_3:attr 5
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
attr a1 is monotone means
(a1 is not non-decreasing) implies a1 is non-increasing;
end;
:: SEQM_3:dfs 6
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is monotone
it is sufficient to prove
thus (a1 is not non-decreasing) implies a1 is non-increasing;
:: SEQM_3:def 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is monotone
iff
(b1 is non-decreasing or b1 is non-increasing);
:: SEQM_3:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is increasing
iff
for b2, b3 being Element of NAT
st b2 < b3
holds b1 . b2 < b1 . b3;
:: SEQM_3:th 8
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is increasing
iff
for b2, b3 being Element of NAT holds
b1 . b2 < b1 . ((b2 + 1) + b3);
:: SEQM_3:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is decreasing
iff
for b2, b3 being Element of NAT holds
b1 . ((b2 + 1) + b3) < b1 . b2;
:: SEQM_3:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is decreasing
iff
for b2, b3 being Element of NAT
st b2 < b3
holds b1 . b3 < b1 . b2;
:: SEQM_3:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is non-decreasing
iff
for b2, b3 being Element of NAT holds
b1 . b2 <= b1 . (b2 + b3);
:: SEQM_3:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is non-decreasing
iff
for b2, b3 being Element of NAT
st b2 <= b3
holds b1 . b2 <= b1 . b3;
:: SEQM_3:th 13
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is non-increasing
iff
for b2, b3 being Element of NAT holds
b1 . (b2 + b3) <= b1 . b2;
:: SEQM_3:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is non-increasing
iff
for b2, b3 being Element of NAT
st b2 <= b3
holds b1 . b3 <= b1 . b2;
:: SEQM_3:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is constant
iff
ex b2 being real set st
proj2 b1 = {b2};
:: SEQM_3:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is constant
iff
for b2 being Element of NAT holds
b1 . b2 = b1 . (b2 + 1);
:: SEQM_3:th 17
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is constant
iff
for b2, b3 being Element of NAT holds
b1 . b2 = b1 . (b2 + b3);
:: SEQM_3:th 18
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is constant
iff
for b2, b3 being Element of NAT holds
b1 . b2 = b1 . b3;
:: SEQM_3:th 19
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is increasing
for b2 being Element of NAT
st 0 < b2
holds b1 . 0 < b1 . b2;
:: SEQM_3:th 20
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is decreasing
for b2 being Element of NAT
st 0 < b2
holds b1 . b2 < b1 . 0;
:: SEQM_3:th 21
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is non-decreasing
for b2 being Element of NAT holds
b1 . 0 <= b1 . b2;
:: SEQM_3:th 22
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is non-increasing
for b2 being Element of NAT holds
b1 . b2 <= b1 . 0;
:: SEQM_3:condreg 1
registration
cluster Function-like increasing -> non-decreasing (Relation of NAT,REAL);
end;
:: SEQM_3:condreg 2
registration
cluster Function-like decreasing -> non-increasing (Relation of NAT,REAL);
end;
:: SEQM_3:condreg 3
registration
cluster Function-like constant -> non-decreasing non-increasing (Relation of NAT,REAL);
end;
:: SEQM_3:condreg 4
registration
cluster Function-like non-decreasing non-increasing -> constant (Relation of NAT,REAL);
end;
:: SEQM_3:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is increasing
holds b1 is non-decreasing;
:: SEQM_3:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is decreasing
holds b1 is non-increasing;
:: SEQM_3:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is constant
holds b1 is non-decreasing;
:: SEQM_3:th 26
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is constant
holds b1 is non-increasing;
:: SEQM_3:th 27
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is non-decreasing & b1 is non-increasing
holds b1 is constant;
:: SEQM_3:attrnot 7 => VALUED_0:attr 6
notation
let a1 be Relation-like set;
synonym natural-yielding for natural-valued;
end;
:: SEQM_3:exreg 1
registration
cluster Relation-like Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing Relation of NAT,REAL;
end;
:: SEQM_3:modenot 1
definition
mode Seq_of_Nat is Function-like quasi_total natural-valued Relation of NAT,REAL;
end;
:: SEQM_3:funcnot 1 => SEQM_3:func 1
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
let a2 be Element of NAT;
func A1 ^\ A2 -> Function-like quasi_total Relation of NAT,REAL means
for b1 being Element of NAT holds
it . b1 = a1 . (b1 + a2);
end;
:: SEQM_3:def 9
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,REAL holds
b3 = b1 ^\ b2
iff
for b4 being Element of NAT holds
b3 . b4 = b1 . (b4 + b2);
:: SEQM_3:th 29
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is Function-like quasi_total natural-valued increasing Relation of NAT,REAL
iff
b1 is increasing &
(for b2 being Element of NAT holds
b1 . b2 is Element of NAT);
:: SEQM_3:th 31
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
for b3 being Element of NAT holds
(b2 * b1) . b3 = b1 . (b2 . b3);
:: SEQM_3:funcnot 2 => SEQM_3:func 2
definition
let a1 be Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
let a2 be Function-like quasi_total Relation of NAT,REAL;
redefine func a2 * a1 -> Function-like quasi_total Relation of NAT,REAL;
end;
:: SEQM_3:funcnot 3 => SEQM_3:func 3
definition
let a1, a2 be Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
redefine func a2 * a1 -> Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
end;
:: SEQM_3:funcreg 1
registration
let a1 be Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
let a2 be Element of NAT;
cluster a1 ^\ a2 -> Function-like quasi_total natural-valued increasing;
end;
:: SEQM_3:modenot 2 => SEQM_3:mode 1
definition
let a1 be Function-like quasi_total Relation of NAT,REAL;
mode subsequence of A1 -> Function-like quasi_total Relation of NAT,REAL means
ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
it = a1 * b1;
end;
:: SEQM_3:dfs 8
definiens
let a1, a2 be Function-like quasi_total Relation of NAT,REAL;
To prove
a2 is subsequence of a1
it is sufficient to prove
thus ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
a2 = a1 * b1;
:: SEQM_3:def 10
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
b2 is subsequence of b1
iff
ex b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
b2 = b1 * b3;
:: SEQM_3:attrnot 8 => SEQM_3:attr 1
definition
let a1 be Function-like Relation of NAT,REAL;
attr a1 is increasing means
for b1 being Element of NAT holds
a1 . b1 < a1 . (b1 + 1);
end;
:: SEQM_3:dfs 9
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is increasing
it is sufficient to prove
thus for b1 being Element of NAT holds
a1 . b1 < a1 . (b1 + 1);
:: SEQM_3:def 11
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is increasing
iff
for b2 being Element of NAT holds
b1 . b2 < b1 . (b2 + 1);
:: SEQM_3:attrnot 9 => SEQM_3:attr 2
definition
let a1 be Function-like Relation of NAT,REAL;
attr a1 is decreasing means
for b1 being Element of NAT holds
a1 . (b1 + 1) < a1 . b1;
end;
:: SEQM_3:dfs 10
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is decreasing
it is sufficient to prove
thus for b1 being Element of NAT holds
a1 . (b1 + 1) < a1 . b1;
:: SEQM_3:def 12
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is decreasing
iff
for b2 being Element of NAT holds
b1 . (b2 + 1) < b1 . b2;
:: SEQM_3:attrnot 10 => SEQM_3:attr 3
definition
let a1 be Function-like Relation of NAT,REAL;
attr a1 is non-decreasing means
for b1 being Element of NAT holds
a1 . b1 <= a1 . (b1 + 1);
end;
:: SEQM_3:dfs 11
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is non-decreasing
it is sufficient to prove
thus for b1 being Element of NAT holds
a1 . b1 <= a1 . (b1 + 1);
:: SEQM_3:def 13
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is non-decreasing
iff
for b2 being Element of NAT holds
b1 . b2 <= b1 . (b2 + 1);
:: SEQM_3:attrnot 11 => SEQM_3:attr 4
definition
let a1 be Function-like Relation of NAT,REAL;
attr a1 is non-increasing means
for b1 being Element of NAT holds
a1 . (b1 + 1) <= a1 . b1;
end;
:: SEQM_3:dfs 12
definiens
let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
a1 is non-increasing
it is sufficient to prove
thus for b1 being Element of NAT holds
a1 . (b1 + 1) <= a1 . b1;
:: SEQM_3:def 14
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is non-increasing
iff
for b2 being Element of NAT holds
b1 . (b2 + 1) <= b1 . b2;
:: SEQM_3:th 33
theorem
for b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
for b2 being Element of NAT holds
b2 <= b1 . b2;
:: SEQM_3:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 ^\ 0 = b1;
:: SEQM_3:th 35
theorem
for b1, b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,REAL holds
(b3 ^\ b1) ^\ b2 = (b3 ^\ b2) ^\ b1;
:: SEQM_3:th 36
theorem
for b1, b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,REAL holds
(b3 ^\ b1) ^\ b2 = b3 ^\ (b1 + b2);
:: SEQM_3:th 37
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b2 + b3) ^\ b1 = (b2 ^\ b1) + (b3 ^\ b1);
:: SEQM_3:th 38
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
(- b2) ^\ b1 = - (b2 ^\ b1);
:: SEQM_3:th 39
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b2 - b3) ^\ b1 = (b2 ^\ b1) - (b3 ^\ b1);
:: SEQM_3:th 40
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is non-empty
holds b2 ^\ b1 is non-empty;
:: SEQM_3:th 41
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
b2 " ^\ b1 = (b2 ^\ b1) ";
:: SEQM_3:th 42
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b2 (#) b3) ^\ b1 = (b2 ^\ b1) (#) (b3 ^\ b1);
:: SEQM_3:th 43
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b2 /" b3) ^\ b1 = (b2 ^\ b1) /" (b3 ^\ b1);
:: SEQM_3:th 44
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL holds
(b2 (#) b3) ^\ b1 = b2 (#) (b3 ^\ b1);
:: SEQM_3:th 45
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
(b2 * b3) ^\ b1 = b2 * (b3 ^\ b1);
:: SEQM_3:th 46
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
b1 is subsequence of b1;
:: SEQM_3:th 47
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
b2 ^\ b1 is subsequence of b2;
:: SEQM_3:th 48
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL
st b1 is subsequence of b2 & b2 is subsequence of b3
holds b1 is subsequence of b3;
:: SEQM_3:th 49
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is increasing & b2 is subsequence of b1
holds b2 is increasing;
:: SEQM_3:th 50
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is decreasing & b2 is subsequence of b1
holds b2 is decreasing;
:: SEQM_3:th 51
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is non-decreasing & b2 is subsequence of b1
holds b2 is non-decreasing;
:: SEQM_3:th 52
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is non-increasing & b2 is subsequence of b1
holds b2 is non-increasing;
:: SEQM_3:th 53
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is monotone & b2 is subsequence of b1
holds b2 is monotone;
:: SEQM_3:th 54
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is constant & b2 is subsequence of b1
holds b2 is constant;
:: SEQM_3:th 55
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is constant & b2 is subsequence of b1
holds b1 = b2;
:: SEQM_3:th 56
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is bounded_above & b2 is subsequence of b1
holds b2 is bounded_above;
:: SEQM_3:th 57
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is bounded_below & b2 is subsequence of b1
holds b2 is bounded_below;
:: SEQM_3:th 58
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is bounded & b2 is subsequence of b1
holds b2 is bounded;
:: SEQM_3:th 59
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
(b2 is increasing & 0 < b1 implies b1 (#) b2 is increasing) &
(0 = b1 implies b1 (#) b2 is constant) &
(b2 is increasing & b1 < 0 implies b1 (#) b2 is decreasing);
:: SEQM_3:th 60
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
(b2 is decreasing & 0 < b1 implies b1 (#) b2 is decreasing) &
(b2 is decreasing & b1 < 0 implies b1 (#) b2 is increasing);
:: SEQM_3:th 61
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
(b2 is non-decreasing & 0 <= b1 implies b1 (#) b2 is non-decreasing) &
(b2 is non-decreasing & b1 <= 0 implies b1 (#) b2 is non-increasing);
:: SEQM_3:th 62
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
(b2 is non-increasing & 0 <= b1 implies b1 (#) b2 is non-increasing) &
(b2 is non-increasing & b1 <= 0 implies b1 (#) b2 is non-decreasing);
:: SEQM_3:th 63
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is increasing & b2 is increasing implies b1 + b2 is increasing) &
(b1 is decreasing & b2 is decreasing implies b1 + b2 is decreasing) &
(b1 is non-decreasing & b2 is non-decreasing implies b1 + b2 is non-decreasing) &
(b1 is non-increasing & b2 is non-increasing implies b1 + b2 is non-increasing);
:: SEQM_3:th 64
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is increasing & b2 is constant implies b1 + b2 is increasing) &
(b1 is decreasing & b2 is constant implies b1 + b2 is decreasing) &
(b1 is non-decreasing & b2 is constant implies b1 + b2 is non-decreasing) &
(b1 is non-increasing & b2 is constant implies b1 + b2 is non-increasing);
:: SEQM_3:th 65
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is constant
holds (for b2 being real set holds
b2 (#) b1 is constant) &
- b1 is constant &
abs b1 is constant;
:: SEQM_3:th 66
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is constant & b2 is constant
holds b1 (#) b2 is constant & b1 + b2 is constant;
:: SEQM_3:th 67
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is constant & b2 is constant
holds b1 - b2 is constant;
:: SEQM_3:th 68
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
(b2 is bounded_above & 0 < b1 implies b1 (#) b2 is bounded_above) &
(0 = b1 implies b1 (#) b2 is bounded) &
(b2 is bounded_above & b1 < 0 implies b1 (#) b2 is bounded_below);
:: SEQM_3:th 69
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
(b2 is bounded_below & 0 < b1 implies b1 (#) b2 is bounded_below) &
(b2 is bounded_below & b1 < 0 implies b1 (#) b2 is bounded_above);
:: SEQM_3:th 70
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is bounded implies for b2 being real set holds
b2 (#) b1 is bounded) &
(b1 is bounded implies - b1 is bounded) &
(b1 is bounded implies abs b1 is bounded) &
(abs b1 is bounded implies b1 is bounded);
:: SEQM_3:th 71
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is bounded_above & b2 is bounded_above implies b1 + b2 is bounded_above) &
(b1 is bounded_below & b2 is bounded_below implies b1 + b2 is bounded_below) &
(b1 is bounded & b2 is bounded implies b1 + b2 is bounded);
:: SEQM_3:th 72
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is bounded & b2 is bounded
holds b1 (#) b2 is bounded & b1 - b2 is bounded;
:: SEQM_3:th 73
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is constant
holds b1 is bounded;
:: SEQM_3:th 74
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is constant
holds (for b2 being real set holds
b2 (#) b1 is bounded) &
- b1 is bounded &
abs b1 is bounded;
:: SEQM_3:th 75
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is bounded_above & b2 is constant implies b1 + b2 is bounded_above) &
(b1 is bounded_below & b2 is constant implies b1 + b2 is bounded_below) &
(b1 is bounded & b2 is constant implies b1 + b2 is bounded);
:: SEQM_3:th 76
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is bounded_above & b2 is constant implies b1 - b2 is bounded_above) &
(b1 is bounded_below & b2 is constant implies b1 - b2 is bounded_below) &
(b1 is bounded & b2 is constant implies b1 - b2 is bounded & b2 - b1 is bounded & b1 (#) b2 is bounded);
:: SEQM_3:th 77
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is bounded_above & b2 is non-increasing
holds b1 + b2 is bounded_above;
:: SEQM_3:th 78
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
st b1 is bounded_below & b2 is non-decreasing
holds b1 + b2 is bounded_below;
:: SEQM_3:th 79
theorem
for b1, b2 being set holds
b1 --> b2 is constant;
:: SEQM_3:funcreg 2
registration
let a1, a2 be set;
cluster a1 --> a2 -> constant;
end;
:: SEQM_3:th 80
theorem
incl NAT is increasing & incl NAT is natural-valued;
:: SEQM_3:condreg 5
registration
cluster -> natural-valued (FinSequence of NAT);
end;