Article MIDSP_2, MML version 4.99.1005
:: MIDSP_2:funcnot 1 => MIDSP_2:func 1
definition
let a1 be non empty addLoopStr;
let a2 be Element of the carrier of a1;
func Double A2 -> Element of the carrier of a1 equals
a2 + a2;
end;
:: MIDSP_2:def 1
theorem
for b1 being non empty addLoopStr
for b2 being Element of the carrier of b1 holds
Double b2 = b2 + b2;
:: MIDSP_2:prednot 1 => MIDSP_2:pred 1
definition
let a1 be non empty MidStr;
let a2 be non empty addLoopStr;
let a3 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of a2;
pred A1,A2 are_associated_wrp A3 means
for b1, b2, b3 being Element of the carrier of a1 holds
b1 @ b2 = b3
iff
a3 .(b1,b3) = a3 .(b3,b2);
end;
:: MIDSP_2:dfs 2
definiens
let a1 be non empty MidStr;
let a2 be non empty addLoopStr;
let a3 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of a2;
To prove
a1,a2 are_associated_wrp a3
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1 holds
b1 @ b2 = b3
iff
a3 .(b1,b3) = a3 .(b3,b2);
:: MIDSP_2:def 2
theorem
for b1 being non empty MidStr
for b2 being non empty addLoopStr
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of b2 holds
b1,b2 are_associated_wrp b3
iff
for b4, b5, b6 being Element of the carrier of b1 holds
b4 @ b5 = b6
iff
b3 .(b4,b6) = b3 .(b6,b5);
:: MIDSP_2:th 1
theorem
for b1 being non empty addLoopStr
for b2 being non empty MidStr
for b3 being Element of the carrier of b2
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1
st b2,b1 are_associated_wrp b4
holds b3 @ b3 = b3;
:: MIDSP_2:prednot 2 => MIDSP_2:pred 2
definition
let a1 be non empty set;
let a2 be non empty addLoopStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
pred A3 is_atlas_of A1,A2 means
(for b1 being Element of a1
for b2 being Element of the carrier of a2 holds
ex b3 being Element of a1 st
a3 .(b1,b3) = b2) &
(for b1, b2, b3 being Element of a1
st a3 .(b1,b2) = a3 .(b1,b3)
holds b2 = b3) &
(for b1, b2, b3 being Element of a1 holds
(a3 .(b1,b2)) + (a3 .(b2,b3)) = a3 .(b1,b3));
end;
:: MIDSP_2:dfs 3
definiens
let a1 be non empty set;
let a2 be non empty addLoopStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
To prove
a3 is_atlas_of a1,a2
it is sufficient to prove
thus (for b1 being Element of a1
for b2 being Element of the carrier of a2 holds
ex b3 being Element of a1 st
a3 .(b1,b3) = b2) &
(for b1, b2, b3 being Element of a1
st a3 .(b1,b2) = a3 .(b1,b3)
holds b2 = b3) &
(for b1, b2, b3 being Element of a1 holds
(a3 .(b1,b2)) + (a3 .(b2,b3)) = a3 .(b1,b3));
:: MIDSP_2:def 3
theorem
for b1 being non empty set
for b2 being non empty addLoopStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2 holds
b3 is_atlas_of b1,b2
iff
(for b4 being Element of b1
for b5 being Element of the carrier of b2 holds
ex b6 being Element of b1 st
b3 .(b4,b6) = b5) &
(for b4, b5, b6 being Element of b1
st b3 .(b4,b5) = b3 .(b4,b6)
holds b5 = b6) &
(for b4, b5, b6 being Element of b1 holds
(b3 .(b4,b5)) + (b3 .(b5,b6)) = b3 .(b4,b6));
:: MIDSP_2:funcnot 2 => MIDSP_2:func 2
definition
let a1 be non empty set;
let a2 be non empty addLoopStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
let a4 be Element of a1;
let a5 be Element of the carrier of a2;
assume a3 is_atlas_of a1,a2;
func (A4,A5). A3 -> Element of a1 means
a3 .(a4,it) = a5;
end;
:: MIDSP_2:def 4
theorem
for b1 being non empty set
for b2 being non empty addLoopStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being Element of b1
for b5 being Element of the carrier of b2
st b3 is_atlas_of b1,b2
for b6 being Element of b1 holds
b6 = (b4,b5). b3
iff
b3 .(b4,b6) = b5;
:: MIDSP_2:th 4
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty right_complementable add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b3
st b4 is_atlas_of b1,b3
holds b4 .(b2,b2) = 0. b3;
:: MIDSP_2:th 5
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being non empty right_complementable add-associative right_zeroed addLoopStr
for b5 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b4
st b5 is_atlas_of b1,b4 & b5 .(b2,b3) = 0. b4
holds b2 = b3;
:: MIDSP_2:th 6
theorem
for b1 being non empty set
for b2, b3 being Element of b1
for b4 being non empty right_complementable add-associative right_zeroed addLoopStr
for b5 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b4
st b5 is_atlas_of b1,b4
holds b5 .(b2,b3) = - (b5 .(b3,b2));
:: MIDSP_2:th 7
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Element of b1
for b6 being non empty right_complementable add-associative right_zeroed addLoopStr
for b7 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b6
st b7 is_atlas_of b1,b6 & b7 .(b2,b3) = b7 .(b4,b5)
holds b7 .(b3,b2) = b7 .(b5,b4);
:: MIDSP_2:th 8
theorem
for b1 being non empty set
for b2 being non empty right_complementable add-associative right_zeroed addLoopStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b3 is_atlas_of b1,b2
for b4 being Element of b1
for b5 being Element of the carrier of b2 holds
ex b6 being Element of b1 st
b3 .(b6,b4) = b5;
:: MIDSP_2:th 9
theorem
for b1 being non empty set
for b2, b3, b4 being Element of b1
for b5 being non empty right_complementable add-associative right_zeroed addLoopStr
for b6 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b5
st b6 is_atlas_of b1,b5 & b6 .(b2,b3) = b6 .(b4,b3)
holds b2 = b4;
:: MIDSP_2:th 10
theorem
for b1 being non empty MidStr
for b2, b3 being Element of the carrier of b1
for b4 being non empty right_complementable add-associative right_zeroed addLoopStr
for b5 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of b4
st b5 is_atlas_of the carrier of b1,b4 & b1,b4 are_associated_wrp b5
holds b2 @ b3 = b3 @ b2;
:: MIDSP_2:th 11
theorem
for b1 being non empty MidStr
for b2, b3 being Element of the carrier of b1
for b4 being non empty right_complementable add-associative right_zeroed addLoopStr
for b5 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of b4
st b5 is_atlas_of the carrier of b1,b4 & b1,b4 are_associated_wrp b5
holds ex b6 being Element of the carrier of b1 st
b6 @ b2 = b3;
:: MIDSP_2:th 13
theorem
for b1 being non empty Abelian add-associative addLoopStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
(b2 + b3) + (b4 + b5) = (b2 + b4) + (b3 + b5);
:: MIDSP_2:th 14
theorem
for b1 being non empty Abelian add-associative addLoopStr
for b2, b3 being Element of the carrier of b1 holds
Double (b2 + b3) = (Double b2) + Double b3;
:: MIDSP_2:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1 holds
Double - b2 = - Double b2;
:: MIDSP_2:th 16
theorem
for b1 being non empty MidStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of b2
st b3 is_atlas_of the carrier of b1,b2 & b1,b2 are_associated_wrp b3
for b4, b5, b6, b7 being Element of the carrier of b1 holds
b4 @ b5 = b6 @ b7
iff
b3 .(b4,b7) = b3 .(b6,b5);
:: MIDSP_2:th 17
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b3 is_atlas_of b1,b2
for b4, b5, b6, b7, b8 being Element of b1
st b3 .(b4,b5) = b3 .(b5,b7) & b3 .(b4,b6) = b3 .(b6,b8)
holds b3 .(b7,b8) = Double (b3 .(b5,b6));
:: MIDSP_2:funcreg 1
registration
let a1 be non empty MidSp-like MidStr;
cluster vectgroup a1 -> right_complementable Abelian add-associative right_zeroed;
end;
:: MIDSP_2:th 18
theorem
for b1 being non empty MidSp-like MidStr holds
(for b2 being set holds
b2 is Element of the carrier of vectgroup b1
iff
b2 is Vector of b1) &
0. vectgroup b1 = ID b1 &
(for b2, b3 being Element of the carrier of vectgroup b1
for b4, b5 being Vector of b1
st b2 = b4 & b3 = b5
holds b2 + b3 = b4 + b5);
:: MIDSP_2:attrnot 1 => MIDSP_2:attr 1
definition
let a1 be non empty addLoopStr;
attr a1 is midpoint_operator means
(for b1 being Element of the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
Double b2 = b1) &
(for b1 being Element of the carrier of a1
st Double b1 = 0. a1
holds b1 = 0. a1);
end;
:: MIDSP_2:dfs 5
definiens
let a1 be non empty addLoopStr;
To prove
a1 is midpoint_operator
it is sufficient to prove
thus (for b1 being Element of the carrier of a1 holds
ex b2 being Element of the carrier of a1 st
Double b2 = b1) &
(for b1 being Element of the carrier of a1
st Double b1 = 0. a1
holds b1 = 0. a1);
:: MIDSP_2:def 5
theorem
for b1 being non empty addLoopStr holds
b1 is midpoint_operator
iff
(for b2 being Element of the carrier of b1 holds
ex b3 being Element of the carrier of b1 st
Double b3 = b2) &
(for b2 being Element of the carrier of b1
st Double b2 = 0. b1
holds b2 = 0. b1);
:: MIDSP_2:condreg 1
registration
cluster non empty midpoint_operator -> Fanoian (addLoopStr);
end;
:: MIDSP_2:exreg 1
registration
cluster non empty strict right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr;
end;
:: MIDSP_2:th 19
theorem
for b1 being non empty right_complementable Fanoian add-associative right_zeroed addLoopStr
for b2 being Element of the carrier of b1
st b2 = - b2
holds b2 = 0. b1;
:: MIDSP_2:th 20
theorem
for b1 being non empty right_complementable Fanoian Abelian add-associative right_zeroed addLoopStr
for b2, b3 being Element of the carrier of b1
st Double b2 = Double b3
holds b2 = b3;
:: MIDSP_2:funcnot 3 => MIDSP_2:func 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr;
let a2 be Element of the carrier of a1;
func Half A2 -> Element of the carrier of a1 means
Double it = a2;
end;
:: MIDSP_2:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b2, b3 being Element of the carrier of b1 holds
b3 = Half b2
iff
Double b3 = b2;
:: MIDSP_2:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b2, b3 being Element of the carrier of b1 holds
Half 0. b1 = 0. b1 &
Half (b2 + b3) = (Half b2) + Half b3 &
(Half b2 = Half b3 implies b2 = b3) &
Half Double b2 = b2;
:: MIDSP_2:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b2 being non empty MidStr
for b3 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1
st b3 is_atlas_of the carrier of b2,b1 & b2,b1 are_associated_wrp b3
for b4, b5, b6, b7 being Element of the carrier of b2 holds
(b4 @ b5) @ (b6 @ b7) = (b4 @ b6) @ (b5 @ b7);
:: MIDSP_2:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b2 being non empty MidStr
for b3 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1
st b3 is_atlas_of the carrier of b2,b1 & b2,b1 are_associated_wrp b3
holds b2 is non empty MidSp-like MidStr;
:: MIDSP_2:funcreg 2
registration
let a1 be non empty MidSp-like MidStr;
cluster vectgroup a1 -> midpoint_operator;
end;
:: MIDSP_2:funcnot 4 => MIDSP_2:func 4
definition
let a1 be non empty MidSp-like MidStr;
let a2, a3 be Element of the carrier of a1;
func vector(A2,A3) -> Element of the carrier of vectgroup a1 equals
vect(a2,a3);
end;
:: MIDSP_2:def 7
theorem
for b1 being non empty MidSp-like MidStr
for b2, b3 being Element of the carrier of b1 holds
vector(b2,b3) = vect(b2,b3);
:: MIDSP_2:funcnot 5 => MIDSP_2:func 5
definition
let a1 be non empty MidSp-like MidStr;
func vect A1 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of vectgroup a1 means
for b1, b2 being Element of the carrier of a1 holds
it .(b1,b2) = vect(b1,b2);
end;
:: MIDSP_2:def 8
theorem
for b1 being non empty MidSp-like MidStr
for b2 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of vectgroup b1 holds
b2 = vect b1
iff
for b3, b4 being Element of the carrier of b1 holds
b2 .(b3,b4) = vect(b3,b4);
:: MIDSP_2:th 24
theorem
for b1 being non empty MidSp-like MidStr holds
vect b1 is_atlas_of the carrier of b1,vectgroup b1;
:: MIDSP_2:th 25
theorem
for b1 being non empty MidSp-like MidStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
vect(b2,b3) = vect(b4,b5)
iff
b2 @ b5 = b3 @ b4;
:: MIDSP_2:th 26
theorem
for b1 being non empty MidSp-like MidStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2 @ b3 = b4
iff
vect(b2,b4) = vect(b4,b3);
:: MIDSP_2:th 27
theorem
for b1 being non empty MidSp-like MidStr holds
b1,vectgroup b1 are_associated_wrp vect b1;
:: MIDSP_2:funcnot 6 => MIDSP_2:func 6
definition
let a1 be non empty set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
assume a3 is_atlas_of a1,a2;
func @ A3 -> Function-like quasi_total Relation of [:a1,a1:],a1 means
for b1, b2 being Element of a1 holds
a3 .(b1,it .(b1,b2)) = a3 .(it .(b1,b2),b2);
end;
:: MIDSP_2:def 9
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b3 is_atlas_of b1,b2
for b4 being Function-like quasi_total Relation of [:b1,b1:],b1 holds
b4 = @ b3
iff
for b5, b6 being Element of b1 holds
b3 .(b5,b4 .(b5,b6)) = b3 .(b4 .(b5,b6),b6);
:: MIDSP_2:th 28
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b3 is_atlas_of b1,b2
for b4, b5, b6 being Element of b1 holds
(@ b3) .(b4,b5) = b6
iff
b3 .(b4,b6) = b3 .(b6,b5);
:: MIDSP_2:funcreg 3
registration
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of [:a1,a1:],a1;
cluster MidStr(#a1,a2#) -> non empty strict;
end;
:: MIDSP_2:funcnot 7 => MIDSP_2:func 7
definition
let a1 be non empty set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
func Atlas A3 -> Function-like quasi_total Relation of [:the carrier of MidStr(#a1,@ a3#),the carrier of MidStr(#a1,@ a3#):],the carrier of a2 equals
a3;
end;
:: MIDSP_2:def 10
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2 holds
Atlas b3 = b3;
:: MIDSP_2:th 32
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b3 is_atlas_of b1,b2
holds MidStr(#b1,@ b3#),b2 are_associated_wrp Atlas b3;
:: MIDSP_2:funcnot 8 => MIDSP_2:func 8
definition
let a1 be non empty set;
let a2 be non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr;
let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
assume a3 is_atlas_of a1,a2;
func MidSp. A3 -> non empty strict MidSp-like MidStr equals
MidStr(#a1,@ a3#);
end;
:: MIDSP_2:def 11
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
st b3 is_atlas_of b1,b2
holds MidSp. b3 = MidStr(#b1,@ b3#);
:: MIDSP_2:th 33
theorem
for b1 being non empty MidStr holds
b1 is non empty MidSp-like MidStr
iff
ex b2 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr st
ex b3 being Function-like quasi_total Relation of [:the carrier of b1,the carrier of b1:],the carrier of b2 st
b3 is_atlas_of the carrier of b1,b2 & b1,b2 are_associated_wrp b3;
:: MIDSP_2:structnot 1 => MIDSP_2:struct 1
definition
let a1 be non empty MidStr;
struct() AtlasStr(#
algebra -> non empty addLoopStr,
function -> Function-like quasi_total Relation of [:the carrier of A1,the carrier of A1:],the carrier of the algebra of it
#);
end;
:: MIDSP_2:attrnot 2 => MIDSP_2:attr 2
definition
let a1 be non empty MidStr;
let a2 be AtlasStr over a1;
attr a2 is strict;
end;
:: MIDSP_2:exreg 2
registration
let a1 be non empty MidStr;
cluster strict AtlasStr over a1;
end;
:: MIDSP_2:aggrnot 1 => MIDSP_2:aggr 1
definition
let a1 be non empty MidStr;
let a2 be non empty addLoopStr;
let a3 be Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of a2;
aggr AtlasStr(#a2,a3#) -> strict AtlasStr over a1;
end;
:: MIDSP_2:selnot 1 => MIDSP_2:sel 1
definition
let a1 be non empty MidStr;
let a2 be AtlasStr over a1;
sel the algebra of a2 -> non empty addLoopStr;
end;
:: MIDSP_2:selnot 2 => MIDSP_2:sel 2
definition
let a1 be non empty MidStr;
let a2 be AtlasStr over a1;
sel the function of a2 -> Function-like quasi_total Relation of [:the carrier of a1,the carrier of a1:],the carrier of the algebra of a2;
end;
:: MIDSP_2:attrnot 3 => MIDSP_2:attr 3
definition
let a1 be non empty MidStr;
let a2 be AtlasStr over a1;
attr a2 is ATLAS-like means
the algebra of a2 is midpoint_operator & the algebra of a2 is add-associative & the algebra of a2 is right_zeroed & the algebra of a2 is right_complementable & the algebra of a2 is Abelian & a1,the algebra of a2 are_associated_wrp the function of a2 & the function of a2 is_atlas_of the carrier of a1,the algebra of a2;
end;
:: MIDSP_2:dfs 12
definiens
let a1 be non empty MidStr;
let a2 be AtlasStr over a1;
To prove
a2 is ATLAS-like
it is sufficient to prove
thus the algebra of a2 is midpoint_operator & the algebra of a2 is add-associative & the algebra of a2 is right_zeroed & the algebra of a2 is right_complementable & the algebra of a2 is Abelian & a1,the algebra of a2 are_associated_wrp the function of a2 & the function of a2 is_atlas_of the carrier of a1,the algebra of a2;
:: MIDSP_2:def 12
theorem
for b1 being non empty MidStr
for b2 being AtlasStr over b1 holds
b2 is ATLAS-like(b1)
iff
the algebra of b2 is midpoint_operator & the algebra of b2 is add-associative & the algebra of b2 is right_zeroed & the algebra of b2 is right_complementable & the algebra of b2 is Abelian & b1,the algebra of b2 are_associated_wrp the function of b2 & the function of b2 is_atlas_of the carrier of b1,the algebra of b2;
:: MIDSP_2:exreg 3
registration
let a1 be non empty MidSp-like MidStr;
cluster ATLAS-like AtlasStr over a1;
end;
:: MIDSP_2:modenot 1
definition
let a1 be non empty MidSp-like MidStr;
mode ATLAS of a1 is ATLAS-like AtlasStr over a1;
end;
:: MIDSP_2:modenot 2
definition
let a1 be non empty MidStr;
let a2 be AtlasStr over a1;
mode Vector of a2 is Element of the carrier of the algebra of a2;
end;
:: MIDSP_2:funcnot 9 => MIDSP_2:func 9
definition
let a1 be non empty MidSp-like MidStr;
let a2 be AtlasStr over a1;
let a3, a4 be Element of the carrier of a1;
func A2 .(A3,A4) -> Element of the carrier of the algebra of a2 equals
(the function of a2) .(a3,a4);
end;
:: MIDSP_2:def 13
theorem
for b1 being non empty MidSp-like MidStr
for b2 being AtlasStr over b1
for b3, b4 being Element of the carrier of b1 holds
b2 .(b3,b4) = (the function of b2) .(b3,b4);
:: MIDSP_2:funcnot 10 => MIDSP_2:func 10
definition
let a1 be non empty MidSp-like MidStr;
let a2 be AtlasStr over a1;
let a3 be Element of the carrier of a1;
let a4 be Element of the carrier of the algebra of a2;
func (A3,A4). A2 -> Element of the carrier of a1 equals
(a3,a4). the function of a2;
end;
:: MIDSP_2:def 14
theorem
for b1 being non empty MidSp-like MidStr
for b2 being AtlasStr over b1
for b3 being Element of the carrier of b1
for b4 being Element of the carrier of the algebra of b2 holds
(b3,b4). b2 = (b3,b4). the function of b2;
:: MIDSP_2:funcnot 11 => MIDSP_2:func 11
definition
let a1 be non empty MidSp-like MidStr;
let a2 be ATLAS-like AtlasStr over a1;
func 0. A2 -> Element of the carrier of the algebra of a2 equals
0. the algebra of a2;
end;
:: MIDSP_2:def 15
theorem
for b1 being non empty MidSp-like MidStr
for b2 being ATLAS-like AtlasStr over b1 holds
0. b2 = 0. the algebra of b2;
:: MIDSP_2:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b2 being non empty MidStr
for b3 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1
for b4, b5, b6, b7 being Element of the carrier of b2
st b3 is_atlas_of the carrier of b2,b1 & b2,b1 are_associated_wrp b3
holds b4 @ b5 = b6 @ b7
iff
b3 .(b4,b5) = (b3 .(b4,b6)) + (b3 .(b4,b7));
:: MIDSP_2:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr
for b2 being non empty MidStr
for b3 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1
for b4, b5, b6 being Element of the carrier of b2
st b3 is_atlas_of the carrier of b2,b1 & b2,b1 are_associated_wrp b3
holds b4 @ b5 = b6
iff
b3 .(b4,b5) = Double (b3 .(b4,b6));
:: MIDSP_2:th 36
theorem
for b1 being non empty MidSp-like MidStr
for b2 being ATLAS-like AtlasStr over b1
for b3, b4, b5, b6 being Element of the carrier of b1 holds
b3 @ b4 = b5 @ b6
iff
b2 .(b3,b4) = (b2 .(b3,b5)) + (b2 .(b3,b6));
:: MIDSP_2:th 37
theorem
for b1 being non empty MidSp-like MidStr
for b2 being ATLAS-like AtlasStr over b1
for b3, b4, b5 being Element of the carrier of b1 holds
b3 @ b4 = b5
iff
b2 .(b3,b4) = Double (b2 .(b3,b5));
:: MIDSP_2:th 38
theorem
for b1 being non empty MidSp-like MidStr
for b2 being ATLAS-like AtlasStr over b1 holds
(for b3 being Element of the carrier of b1
for b4 being Element of the carrier of the algebra of b2 holds
ex b5 being Element of the carrier of b1 st
b2 .(b3,b5) = b4) &
(for b3, b4, b5 being Element of the carrier of b1
st b2 .(b3,b4) = b2 .(b3,b5)
holds b4 = b5) &
(for b3, b4, b5 being Element of the carrier of b1 holds
(b2 .(b3,b4)) + (b2 .(b4,b5)) = b2 .(b3,b5));
:: MIDSP_2:th 39
theorem
for b1 being non empty MidSp-like MidStr
for b2 being ATLAS-like AtlasStr over b1
for b3, b4, b5, b6 being Element of the carrier of b1
for b7 being Element of the carrier of the algebra of b2 holds
b2 .(b3,b3) = 0. b2 &
(b2 .(b3,b4) = 0. b2 implies b3 = b4) &
b2 .(b3,b4) = - (b2 .(b4,b3)) &
(b2 .(b3,b4) = b2 .(b5,b6) implies b2 .(b4,b3) = b2 .(b6,b5)) &
(for b8 being Element of the carrier of b1
for b9 being Element of the carrier of the algebra of b2 holds
ex b10 being Element of the carrier of b1 st
b2 .(b10,b8) = b9) &
(b2 .(b4,b3) = b2 .(b5,b3) implies b4 = b5) &
(b3 @ b4 = b5 implies b2 .(b3,b5) = b2 .(b5,b4)) &
(b2 .(b3,b5) = b2 .(b5,b4) implies b3 @ b4 = b5) &
(b3 @ b4 = b5 @ b6 implies b2 .(b3,b6) = b2 .(b5,b4)) &
(b2 .(b3,b6) = b2 .(b5,b4) implies b3 @ b4 = b5 @ b6) &
(b2 .(b3,b4) = b7 implies (b3,b7). b2 = b4) &
((b3,b7). b2 = b4 implies b2 .(b3,b4) = b7);
:: MIDSP_2:th 40
theorem
for b1 being non empty MidSp-like MidStr
for b2 being ATLAS-like AtlasStr over b1
for b3 being Element of the carrier of b1 holds
(b3,0. b2). b2 = b3;