Article ARYTM_3, MML version 4.99.1005
:: ARYTM_3:funcnot 1 => ARYTM_3:func 1
definition
func one -> set equals
1;
end;
:: ARYTM_3:def 1
theorem
one = 1;
:: ARYTM_3:prednot 1 => ARYTM_3:pred 1
definition
let a1, a2 be ordinal set;
pred A1,A2 are_relative_prime means
for b1, b2, b3 being ordinal set
st a1 = b1 *^ b2 & a2 = b1 *^ b3
holds b1 = 1;
symmetry;
:: for a1, a2 being ordinal set
:: st a1,a2 are_relative_prime
:: holds a2,a1 are_relative_prime;
end;
:: ARYTM_3:dfs 2
definiens
let a1, a2 be ordinal set;
To prove
a1,a2 are_relative_prime
it is sufficient to prove
thus for b1, b2, b3 being ordinal set
st a1 = b1 *^ b2 & a2 = b1 *^ b3
holds b1 = 1;
:: ARYTM_3:def 2
theorem
for b1, b2 being ordinal set holds
b1,b2 are_relative_prime
iff
for b3, b4, b5 being ordinal set
st b1 = b3 *^ b4 & b2 = b3 *^ b5
holds b3 = 1;
:: ARYTM_3:th 5
theorem
not {},{} are_relative_prime;
:: ARYTM_3:th 6
theorem
for b1 being ordinal set holds
1,b1 are_relative_prime;
:: ARYTM_3:th 7
theorem
for b1 being ordinal set
st {},b1 are_relative_prime
holds b1 = 1;
:: ARYTM_3:th 8
theorem
for b1, b2 being ordinal natural set
st (b1 = {} implies b2 <> {})
holds ex b3, b4, b5 being ordinal natural set st
b4,b5 are_relative_prime & b1 = b3 *^ b4 & b2 = b3 *^ b5;
:: ARYTM_3:funcreg 1
registration
let a1, a2 be ordinal natural set;
cluster a1 div^ a2 -> ordinal natural;
end;
:: ARYTM_3:funcreg 2
registration
let a1, a2 be ordinal natural set;
cluster a1 mod^ a2 -> ordinal natural;
end;
:: ARYTM_3:prednot 2 => ARYTM_3:pred 2
definition
let a1, a2 be ordinal set;
pred A1 divides A2 means
ex b1 being ordinal set st
a2 = a1 *^ b1;
reflexivity;
:: for a1 being ordinal set holds
:: a1 divides a1;
end;
:: ARYTM_3:dfs 3
definiens
let a1, a2 be ordinal set;
To prove
a1 divides a2
it is sufficient to prove
thus ex b1 being ordinal set st
a2 = a1 *^ b1;
:: ARYTM_3:def 3
theorem
for b1, b2 being ordinal set holds
b1 divides b2
iff
ex b3 being ordinal set st
b2 = b1 *^ b3;
:: ARYTM_3:th 9
theorem
for b1, b2 being ordinal natural set holds
b1 divides b2
iff
ex b3 being ordinal natural set st
b2 = b1 *^ b3;
:: ARYTM_3:th 10
theorem
for b1, b2 being ordinal natural set
st {} in b1
holds b2 mod^ b1 in b1;
:: ARYTM_3:th 11
theorem
for b1, b2 being ordinal natural set holds
b2 divides b1
iff
b1 = b2 *^ (b1 div^ b2);
:: ARYTM_3:th 13
theorem
for b1, b2 being ordinal natural set
st b1 divides b2 & b2 divides b1
holds b1 = b2;
:: ARYTM_3:th 14
theorem
for b1 being ordinal natural set holds
b1 divides {} & 1 divides b1;
:: ARYTM_3:th 15
theorem
for b1, b2 being ordinal natural set
st {} in b2 & b1 divides b2
holds b1 c= b2;
:: ARYTM_3:th 16
theorem
for b1, b2, b3 being ordinal natural set
st b1 divides b2 & b1 divides b2 +^ b3
holds b1 divides b3;
:: ARYTM_3:funcnot 2 => ARYTM_3:func 2
definition
let a1, a2 be ordinal natural set;
func A1 lcm A2 -> Element of omega means
a1 divides it &
a2 divides it &
(for b1 being ordinal natural set
st a1 divides b1 & a2 divides b1
holds it divides b1);
commutativity;
:: for a1, a2 being ordinal natural set holds
:: a1 lcm a2 = a2 lcm a1;
end;
:: ARYTM_3:def 4
theorem
for b1, b2 being ordinal natural set
for b3 being Element of omega holds
b3 = b1 lcm b2
iff
b1 divides b3 &
b2 divides b3 &
(for b4 being ordinal natural set
st b1 divides b4 & b2 divides b4
holds b3 divides b4);
:: ARYTM_3:th 17
theorem
for b1, b2 being ordinal natural set holds
b1 lcm b2 divides b1 *^ b2;
:: ARYTM_3:th 18
theorem
for b1, b2 being ordinal natural set
st b1 <> {}
holds (b2 *^ b1) div^ (b2 lcm b1) divides b2;
:: ARYTM_3:funcnot 3 => ARYTM_3:func 3
definition
let a1, a2 be ordinal natural set;
func A1 hcf A2 -> Element of omega means
it divides a1 &
it divides a2 &
(for b1 being ordinal natural set
st b1 divides a1 & b1 divides a2
holds b1 divides it);
commutativity;
:: for a1, a2 being ordinal natural set holds
:: a1 hcf a2 = a2 hcf a1;
end;
:: ARYTM_3:def 5
theorem
for b1, b2 being ordinal natural set
for b3 being Element of omega holds
b3 = b1 hcf b2
iff
b3 divides b1 &
b3 divides b2 &
(for b4 being ordinal natural set
st b4 divides b1 & b4 divides b2
holds b4 divides b3);
:: ARYTM_3:th 19
theorem
for b1 being ordinal natural set holds
b1 hcf {} = b1 & b1 lcm {} = {};
:: ARYTM_3:th 20
theorem
for b1, b2 being ordinal natural set
st b1 hcf b2 = {}
holds b1 = {};
:: ARYTM_3:th 21
theorem
for b1 being ordinal natural set holds
b1 hcf b1 = b1 & b1 lcm b1 = b1;
:: ARYTM_3:th 22
theorem
for b1, b2, b3 being ordinal natural set holds
(b1 *^ b2) hcf (b3 *^ b2) = (b1 hcf b3) *^ b2;
:: ARYTM_3:th 23
theorem
for b1, b2 being ordinal natural set
st b1 <> {}
holds b2 hcf b1 <> {} & b1 div^ (b2 hcf b1) <> {};
:: ARYTM_3:th 24
theorem
for b1, b2 being ordinal natural set
st (b1 = {} implies b2 <> {})
holds b1 div^ (b1 hcf b2),b2 div^ (b1 hcf b2) are_relative_prime;
:: ARYTM_3:th 25
theorem
for b1, b2 being ordinal natural set holds
b1,b2 are_relative_prime
iff
b1 hcf b2 = 1;
:: ARYTM_3:funcnot 4 => ARYTM_3:func 4
definition
let a1, a2 be ordinal natural set;
func RED(A1,A2) -> Element of omega equals
a1 div^ (a1 hcf a2);
end;
:: ARYTM_3:def 6
theorem
for b1, b2 being ordinal natural set holds
RED(b1,b2) = b1 div^ (b1 hcf b2);
:: ARYTM_3:th 26
theorem
for b1, b2 being ordinal natural set holds
(RED(b1,b2)) *^ (b1 hcf b2) = b1;
:: ARYTM_3:th 27
theorem
for b1, b2 being ordinal natural set
st (b1 = {} implies b2 <> {})
holds RED(b1,b2),RED(b2,b1) are_relative_prime;
:: ARYTM_3:th 28
theorem
for b1, b2 being ordinal natural set
st b1,b2 are_relative_prime
holds RED(b1,b2) = b1;
:: ARYTM_3:th 29
theorem
for b1 being ordinal natural set holds
RED(b1,1) = b1 & RED(1,b1) = 1;
:: ARYTM_3:th 30
theorem
for b1, b2 being ordinal natural set
st b1 <> {}
holds RED(b1,b2) <> {};
:: ARYTM_3:th 31
theorem
for b1 being ordinal natural set holds
RED({},b1) = {} & (b1 = {} or RED(b1,{}) = 1);
:: ARYTM_3:th 32
theorem
for b1 being ordinal natural set
st b1 <> {}
holds RED(b1,b1) = 1;
:: ARYTM_3:th 33
theorem
for b1, b2, b3 being ordinal natural set
st b1 <> {}
holds RED(b2 *^ b1,b3 *^ b1) = RED(b2,b3);
:: ARYTM_3:funcnot 5 => ARYTM_3:func 5
definition
func RAT+ -> set equals
({[b1,b2] where b1 is Element of omega, b2 is Element of omega: b1,b2 are_relative_prime & b2 <> {}} \ {[b1,1] where b1 is Element of omega: TRUE}) \/ omega;
end;
:: ARYTM_3:def 7
theorem
RAT+ = ({[b1,b2] where b1 is Element of omega, b2 is Element of omega: b1,b2 are_relative_prime & b2 <> {}} \ {[b1,1] where b1 is Element of omega: TRUE}) \/ omega;
:: ARYTM_3:funcreg 3
registration
cluster RAT+ -> non empty;
end;
:: ARYTM_3:exreg 1
registration
cluster non empty ordinal Element of RAT+;
end;
:: ARYTM_3:th 35
theorem
for b1 being Element of RAT+
st not b1 in omega
holds ex b2, b3 being Element of omega st
b1 = [b2,b3] & b2,b3 are_relative_prime & b3 <> {} & b3 <> 1;
:: ARYTM_3:th 36
theorem
for b1, b2 being set holds
[b1,b2] is not ordinal set;
:: ARYTM_3:th 37
theorem
for b1 being ordinal set
st b1 in RAT+
holds b1 in omega;
:: ARYTM_3:condreg 1
registration
cluster ordinal -> natural (Element of RAT+);
end;
:: ARYTM_3:th 38
theorem
for b1, b2 being set holds
not [b1,b2] in omega;
:: ARYTM_3:th 39
theorem
for b1, b2 being Element of omega holds
[b1,b2] in RAT+
iff
b1,b2 are_relative_prime & b2 <> {} & b2 <> 1;
:: ARYTM_3:funcnot 6 => ARYTM_3:func 6
definition
let a1 be Element of RAT+;
func numerator A1 -> Element of omega means
it = a1
if a1 in omega
otherwise ex b1 being ordinal natural set st
a1 = [it,b1];
end;
:: ARYTM_3:def 8
theorem
for b1 being Element of RAT+
for b2 being Element of omega holds
(b1 in omega implies (b2 = numerator b1
iff
b2 = b1)) &
(b1 in omega or (b2 = numerator b1
iff
ex b3 being ordinal natural set st
b1 = [b2,b3]));
:: ARYTM_3:funcnot 7 => ARYTM_3:func 7
definition
let a1 be Element of RAT+;
func denominator A1 -> Element of omega means
it = 1
if a1 in omega
otherwise ex b1 being ordinal natural set st
a1 = [b1,it];
end;
:: ARYTM_3:def 9
theorem
for b1 being Element of RAT+
for b2 being Element of omega holds
(b1 in omega implies (b2 = denominator b1
iff
b2 = 1)) &
(b1 in omega or (b2 = denominator b1
iff
ex b3 being ordinal natural set st
b1 = [b3,b2]));
:: ARYTM_3:th 40
theorem
for b1 being Element of RAT+ holds
numerator b1,denominator b1 are_relative_prime;
:: ARYTM_3:th 41
theorem
for b1 being Element of RAT+ holds
denominator b1 <> {};
:: ARYTM_3:th 42
theorem
for b1 being Element of RAT+
st not b1 in omega
holds b1 = [numerator b1,denominator b1] & denominator b1 <> 1;
:: ARYTM_3:th 43
theorem
for b1 being Element of RAT+ holds
b1 <> {}
iff
numerator b1 <> {};
:: ARYTM_3:th 44
theorem
for b1 being Element of RAT+ holds
b1 in omega
iff
denominator b1 = 1;
:: ARYTM_3:funcnot 8 => ARYTM_3:func 8
definition
let a1, a2 be ordinal natural set;
func A1 / A2 -> Element of RAT+ equals
{}
if a2 = {},
RED(a1,a2)
if RED(a2,a1) = 1
otherwise [RED(a1,a2),RED(a2,a1)];
end;
:: ARYTM_3:def 10
theorem
for b1, b2 being ordinal natural set holds
(b2 = {} implies b1 / b2 = {}) &
(RED(b2,b1) = 1 implies b1 / b2 = RED(b1,b2)) &
(b2 <> {} & RED(b2,b1) <> 1 implies b1 / b2 = [RED(b1,b2),RED(b2,b1)]);
:: ARYTM_3:funcnot 9 => ARYTM_3:func 8
notation
let a1, a2 be ordinal natural set;
synonym quotient(a1,a2) for a1 / a2;
end;
:: ARYTM_3:th 45
theorem
for b1 being Element of RAT+ holds
(numerator b1) / denominator b1 = b1;
:: ARYTM_3:th 46
theorem
for b1, b2 being ordinal natural set holds
{} / b1 = {} & b2 / 1 = b2;
:: ARYTM_3:th 47
theorem
for b1 being ordinal natural set
st b1 <> {}
holds b1 / b1 = 1;
:: ARYTM_3:th 48
theorem
for b1, b2 being ordinal natural set
st b1 <> {}
holds numerator (b2 / b1) = RED(b2,b1) & denominator (b2 / b1) = RED(b1,b2);
:: ARYTM_3:th 49
theorem
for b1, b2 being Element of omega
st b1,b2 are_relative_prime & b2 <> {}
holds numerator (b1 / b2) = b1 & denominator (b1 / b2) = b2;
:: ARYTM_3:th 50
theorem
for b1, b2, b3 being ordinal natural set
st b1 <> {}
holds (b2 *^ b1) / (b3 *^ b1) = b2 / b3;
:: ARYTM_3:th 51
theorem
for b1, b2, b3, b4 being ordinal natural set
st b2 <> {} & b1 <> {}
holds b3 / b2 = b4 / b1
iff
b3 *^ b1 = b2 *^ b4;
:: ARYTM_3:funcnot 10 => ARYTM_3:func 9
definition
let a1, a2 be Element of RAT+;
func A1 + A2 -> Element of RAT+ equals
(((numerator a1) *^ denominator a2) +^ ((numerator a2) *^ denominator a1)) / ((denominator a1) *^ denominator a2);
commutativity;
:: for a1, a2 being Element of RAT+ holds
:: a1 + a2 = a2 + a1;
end;
:: ARYTM_3:def 11
theorem
for b1, b2 being Element of RAT+ holds
b1 + b2 = (((numerator b1) *^ denominator b2) +^ ((numerator b2) *^ denominator b1)) / ((denominator b1) *^ denominator b2);
:: ARYTM_3:funcnot 11 => ARYTM_3:func 10
definition
let a1, a2 be Element of RAT+;
func A1 *' A2 -> Element of RAT+ equals
((numerator a1) *^ numerator a2) / ((denominator a1) *^ denominator a2);
commutativity;
:: for a1, a2 being Element of RAT+ holds
:: a1 *' a2 = a2 *' a1;
end;
:: ARYTM_3:def 12
theorem
for b1, b2 being Element of RAT+ holds
b1 *' b2 = ((numerator b1) *^ numerator b2) / ((denominator b1) *^ denominator b2);
:: ARYTM_3:th 52
theorem
for b1, b2, b3, b4 being ordinal natural set
st b2 <> {} & b1 <> {}
holds (b3 / b2) + (b4 / b1) = ((b3 *^ b1) +^ (b2 *^ b4)) / (b2 *^ b1);
:: ARYTM_3:th 53
theorem
for b1, b2, b3 being ordinal natural set
st b1 <> {}
holds (b2 / b1) + (b3 / b1) = (b2 +^ b3) / b1;
:: ARYTM_3:exreg 2
registration
cluster empty Element of RAT+;
end;
:: ARYTM_3:funcnot 12 => ARYTM_3:func 11
definition
redefine func {} -> Element of RAT+;
end;
:: ARYTM_3:funcnot 13 => ARYTM_3:func 12
definition
redefine func one -> non empty ordinal Element of RAT+;
end;
:: ARYTM_3:th 54
theorem
for b1 being Element of RAT+ holds
b1 *' {} = {};
:: ARYTM_3:th 55
theorem
for b1, b2, b3, b4 being ordinal natural set holds
(b2 / b3) *' (b4 / b1) = (b2 *^ b4) / (b3 *^ b1);
:: ARYTM_3:th 56
theorem
for b1 being Element of RAT+ holds
b1 + {} = b1;
:: ARYTM_3:th 57
theorem
for b1, b2, b3 being Element of RAT+ holds
(b1 + b2) + b3 = b1 + (b2 + b3);
:: ARYTM_3:th 58
theorem
for b1, b2, b3 being Element of RAT+ holds
(b1 *' b2) *' b3 = b1 *' (b2 *' b3);
:: ARYTM_3:th 59
theorem
for b1 being Element of RAT+ holds
b1 *' one = b1;
:: ARYTM_3:th 60
theorem
for b1 being Element of RAT+
st b1 <> {}
holds ex b2 being Element of RAT+ st
b1 *' b2 = 1;
:: ARYTM_3:th 61
theorem
for b1, b2 being Element of RAT+
st b1 <> {}
holds ex b3 being Element of RAT+ st
b2 = b1 *' b3;
:: ARYTM_3:th 62
theorem
for b1, b2, b3 being Element of RAT+
st b1 <> {} & b1 *' b2 = b1 *' b3
holds b2 = b3;
:: ARYTM_3:th 63
theorem
for b1, b2, b3 being Element of RAT+ holds
b1 *' (b2 + b3) = (b1 *' b2) + (b1 *' b3);
:: ARYTM_3:th 64
theorem
for b1, b2 being ordinal Element of RAT+ holds
b1 + b2 = b1 +^ b2;
:: ARYTM_3:th 65
theorem
for b1, b2 being ordinal Element of RAT+ holds
b1 *' b2 = b1 *^ b2;
:: ARYTM_3:th 66
theorem
for b1 being Element of RAT+ holds
ex b2 being Element of RAT+ st
b1 = b2 + b2;
:: ARYTM_3:prednot 3 => ARYTM_3:pred 3
definition
let a1, a2 be Element of RAT+;
pred A1 <=' A2 means
ex b1 being Element of RAT+ st
a2 = a1 + b1;
connectedness;
:: for a1, a2 being Element of RAT+
:: st a2 < a1
:: holds a2 <=' a1;
end;
:: ARYTM_3:dfs 13
definiens
let a1, a2 be Element of RAT+;
To prove
a1 <=' a2
it is sufficient to prove
thus ex b1 being Element of RAT+ st
a2 = a1 + b1;
:: ARYTM_3:def 13
theorem
for b1, b2 being Element of RAT+ holds
b1 <=' b2
iff
ex b3 being Element of RAT+ st
b2 = b1 + b3;
:: ARYTM_3:prednot 4 => not ARYTM_3:pred 3
notation
let a1, a2 be Element of RAT+;
antonym a2 < a1 for a1 <=' a2;
end;
:: ARYTM_3:th 68
theorem
for b1 being set holds
not [{},b1] in RAT+;
:: ARYTM_3:th 69
theorem
for b1, b2, b3 being Element of RAT+
st b1 + b2 = b3 + b2
holds b1 = b3;
:: ARYTM_3:th 70
theorem
for b1, b2 being Element of RAT+
st b1 + b2 = {}
holds b1 = {};
:: ARYTM_3:th 71
theorem
for b1 being Element of RAT+ holds
{} <=' b1;
:: ARYTM_3:th 72
theorem
for b1 being Element of RAT+
st b1 <=' {}
holds b1 = {};
:: ARYTM_3:th 73
theorem
for b1, b2 being Element of RAT+
st b1 <=' b2 & b2 <=' b1
holds b1 = b2;
:: ARYTM_3:th 74
theorem
for b1, b2, b3 being Element of RAT+
st b1 <=' b2 & b2 <=' b3
holds b1 <=' b3;
:: ARYTM_3:th 75
theorem
for b1, b2 being Element of RAT+ holds
b1 < b2
iff
b1 <=' b2 & b1 <> b2;
:: ARYTM_3:th 76
theorem
for b1, b2, b3 being Element of RAT+
st (b1 < b2 & b2 <=' b3 or b1 <=' b2 & b2 < b3)
holds b1 < b3;
:: ARYTM_3:th 77
theorem
for b1, b2, b3 being Element of RAT+
st b1 < b2 & b2 < b3
holds b1 < b3;
:: ARYTM_3:th 78
theorem
for b1, b2 being Element of RAT+
st b1 in omega & b1 + b2 in omega
holds b2 in omega;
:: ARYTM_3:th 79
theorem
for b1 being Element of RAT+
for b2 being ordinal Element of RAT+
st b2 < b1 & b1 < b2 + one
holds not b1 in omega;
:: ARYTM_3:th 80
theorem
for b1 being Element of RAT+
st b1 <> {}
holds ex b2 being Element of RAT+ st
b2 < b1 & not b2 in omega;
:: ARYTM_3:th 81
theorem
for b1 being Element of RAT+ holds
{b2 where b2 is Element of RAT+: b2 < b1} in RAT+
iff
b1 = {};
:: ARYTM_3:th 82
theorem
for b1 being Element of bool RAT+
st (ex b2 being Element of RAT+ st
b2 in b1 & b2 <> {}) &
(for b2, b3 being Element of RAT+
st b2 in b1 & b3 <=' b2
holds b3 in b1)
holds ex b2, b3, b4 being Element of RAT+ st
b2 in b1 & b3 in b1 & b4 in b1 & b2 <> b3 & b3 <> b4 & b4 <> b2;
:: ARYTM_3:th 83
theorem
for b1, b2, b3 being Element of RAT+ holds
b1 + b2 <=' b3 + b2
iff
b1 <=' b3;
:: ARYTM_3:th 85
theorem
for b1, b2 being Element of RAT+ holds
b1 <=' b1 + b2;
:: ARYTM_3:th 86
theorem
for b1, b2 being Element of RAT+
st b1 *' b2 = {} & b1 <> {}
holds b2 = {};
:: ARYTM_3:th 87
theorem
for b1, b2, b3 being Element of RAT+
st b1 <=' b2 *' b3
holds ex b4 being Element of RAT+ st
b1 = b2 *' b4 & b4 <=' b3;
:: ARYTM_3:th 88
theorem
for b1, b2, b3 being Element of RAT+
st b1 <> {} & b2 *' b1 <=' b3 *' b1
holds b2 <=' b3;
:: ARYTM_3:th 89
theorem
for b1, b2, b3, b4 being Element of RAT+
st b1 + b2 = b3 + b4 & b3 < b1
holds b2 <=' b4;
:: ARYTM_3:th 90
theorem
for b1, b2, b3 being Element of RAT+
st b1 <=' b2
holds b1 *' b3 <=' b2 *' b3;
:: ARYTM_3:th 91
theorem
for b1, b2, b3, b4 being Element of RAT+
st b1 *' b2 = b3 *' b4 & b3 < b1
holds b2 <=' b4;
:: ARYTM_3:th 92
theorem
for b1, b2 being Element of RAT+ holds
b1 = {}
iff
b1 + b2 = b2;
:: ARYTM_3:th 93
theorem
for b1, b2, b3, b4 being Element of RAT+
st b1 + b2 = b3 + b4 & b1 <=' b3
holds b4 <=' b2;
:: ARYTM_3:th 94
theorem
for b1, b2, b3 being Element of RAT+
st b1 <=' b2 & b2 <=' b1 + b3
holds ex b4 being Element of RAT+ st
b2 = b1 + b4 & b4 <=' b3;
:: ARYTM_3:th 95
theorem
for b1, b2, b3 being Element of RAT+
st b1 <=' b2 + b3
holds ex b4, b5 being Element of RAT+ st
b1 = b4 + b5 & b4 <=' b2 & b5 <=' b3;
:: ARYTM_3:th 96
theorem
for b1, b2, b3 being Element of RAT+
st b1 < b2 & b1 < b3
holds ex b4 being Element of RAT+ st
b4 <=' b2 & b4 <=' b3 & b1 < b4;
:: ARYTM_3:th 97
theorem
for b1, b2, b3 being Element of RAT+
st b1 <=' b2 & b2 <=' b3 & b2 <> b3
holds b1 <> b3;
:: ARYTM_3:th 98
theorem
for b1, b2, b3 being Element of RAT+
st b1 < b2 + b3 & b3 <> {}
holds ex b4, b5 being Element of RAT+ st
b1 = b4 + b5 & b4 <=' b2 & b5 <=' b3 & b5 <> b3;
:: ARYTM_3:th 99
theorem
for b1 being non empty Element of bool RAT+
st b1 in RAT+
holds ex b2 being Element of RAT+ st
b2 in b1 &
(for b3 being Element of RAT+
st b3 in b1
holds b3 <=' b2);
:: ARYTM_3:th 100
theorem
for b1, b2 being Element of RAT+ holds
ex b3 being Element of RAT+ st
(b1 + b3 = b2 or b2 + b3 = b1);
:: ARYTM_3:th 101
theorem
for b1, b2 being Element of RAT+
st b1 < b2
holds ex b3 being Element of RAT+ st
b1 < b3 & b3 < b2;
:: ARYTM_3:th 102
theorem
for b1 being Element of RAT+ holds
ex b2 being Element of RAT+ st
b1 < b2;
:: ARYTM_3:th 103
theorem
for b1, b2 being Element of RAT+
st b1 <> {}
holds ex b3 being Element of RAT+ st
b3 in omega & b2 <=' b3 *' b1;
:: ARYTM_3:sch 1
scheme ARYTM_3:sch 1
{F1 -> Element of RAT+,
F2 -> Element of RAT+,
F3 -> Element of RAT+}:
ex b1 being Element of RAT+ st
b1 in omega & P1[b1] & not (P1[b1 + F2()])
provided
F2() = 1
and
F1() = {}
and
F3() in omega
and
P1[F1()]
and
not (P1[F3()]);