Article BILINEAR, MML version 4.99.1005
:: BILINEAR:modenot 1
definition
let a1 be 1-sorted;
let a2, a3 be VectSpStr over a1;
mode Form of a2,a3 is Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
end;
:: BILINEAR:funcnot 1 => BILINEAR:func 1
definition
let a1 be non empty ZeroStr;
let a2, a3 be VectSpStr over a1;
func NulForm(A2,A3) -> Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1 equals
[:the carrier of a2,the carrier of a3:] --> 0. a1;
end;
:: BILINEAR:def 2
theorem
for b1 being non empty ZeroStr
for b2, b3 being VectSpStr over b1 holds
NulForm(b2,b3) = [:the carrier of b2,the carrier of b3:] --> 0. b1;
:: BILINEAR:funcnot 2 => BILINEAR:func 2
definition
let a1 be non empty addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
func A4 + A5 -> Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1 means
for b1 being Element of the carrier of a2
for b2 being Element of the carrier of a3 holds
it .(b1,b2) = (a4 .(b1,b2)) + (a5 .(b1,b2));
end;
:: BILINEAR:def 3
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5, b6 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b6 = b4 + b5
iff
for b7 being Element of the carrier of b2
for b8 being Element of the carrier of b3 holds
b6 .(b7,b8) = (b4 .(b7,b8)) + (b5 .(b7,b8));
:: BILINEAR:funcnot 3 => BILINEAR:func 3
definition
let a1 be non empty multMagma;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a1;
func A5 * A4 -> Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1 means
for b1 being Element of the carrier of a2
for b2 being Element of the carrier of a3 holds
it .(b1,b2) = a5 * (a4 .(b1,b2));
end;
:: BILINEAR:def 4
theorem
for b1 being non empty multMagma
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b1
for b6 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b6 = b5 * b4
iff
for b7 being Element of the carrier of b2
for b8 being Element of the carrier of b3 holds
b6 .(b7,b8) = b5 * (b4 .(b7,b8));
:: BILINEAR:funcnot 4 => BILINEAR:func 4
definition
let a1 be non empty addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
func - A4 -> Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1 means
for b1 being Element of the carrier of a2
for b2 being Element of the carrier of a3 holds
it .(b1,b2) = - (a4 .(b1,b2));
end;
:: BILINEAR:def 5
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b5 = - b4
iff
for b6 being Element of the carrier of b2
for b7 being Element of the carrier of b3 holds
b5 .(b6,b7) = - (b4 .(b6,b7));
:: BILINEAR:funcnot 5 => BILINEAR:func 5
definition
let a1 be non empty right_complementable add-associative right_zeroed left-distributive left_unital doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
redefine func - A4 -> Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1 equals
(- 1. a1) * a4;
end;
:: BILINEAR:def 6
theorem
for b1 being non empty right_complementable add-associative right_zeroed left-distributive left_unital doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
- b4 = (- 1. b1) * b4;
:: BILINEAR:funcnot 6 => BILINEAR:func 6
definition
let a1 be non empty addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
func A4 - A5 -> Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1 equals
a4 + - a5;
end;
:: BILINEAR:def 7
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b4 - b5 = b4 + - b5;
:: BILINEAR:funcnot 7 => BILINEAR:func 7
definition
let a1 be non empty addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
redefine func A4 - A5 -> Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1 means
for b1 being Element of the carrier of a2
for b2 being Element of the carrier of a3 holds
it .(b1,b2) = (a4 .(b1,b2)) - (a5 .(b1,b2));
end;
:: BILINEAR:def 8
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5, b6 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b6 = b4 - b5
iff
for b7 being Element of the carrier of b2
for b8 being Element of the carrier of b3 holds
b6 .(b7,b8) = (b4 .(b7,b8)) - (b5 .(b7,b8));
:: BILINEAR:funcnot 8 => BILINEAR:func 8
definition
let a1 be non empty Abelian addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
redefine func a4 + a5 -> Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
commutativity;
:: for a1 being non empty Abelian addLoopStr
:: for a2, a3 being non empty VectSpStr over a1
:: for a4, a5 being Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1 holds
:: a4 + a5 = a5 + a4;
end;
:: BILINEAR:th 2
theorem
for b1 being non empty right_zeroed addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b4 + NulForm(b2,b3) = b4;
:: BILINEAR:th 3
theorem
for b1 being non empty add-associative addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5, b6 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
(b4 + b5) + b6 = b4 + (b5 + b6);
:: BILINEAR:th 4
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b4 - b4 = NulForm(b2,b3);
:: BILINEAR:th 5
theorem
for b1 being non empty right-distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Element of the carrier of b1
for b5, b6 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b4 * (b5 + b6) = (b4 * b5) + (b4 * b6);
:: BILINEAR:th 6
theorem
for b1 being non empty left-distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5 being Element of the carrier of b1
for b6 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
(b4 + b5) * b6 = (b4 * b6) + (b5 * b6);
:: BILINEAR:th 7
theorem
for b1 being non empty associative doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5 being Element of the carrier of b1
for b6 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
(b4 * b5) * b6 = b4 * (b5 * b6);
:: BILINEAR:th 8
theorem
for b1 being non empty left_unital doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
(1. b1) * b4 = b4;
:: BILINEAR:funcnot 9 => BILINEAR:func 9
definition
let a1 be non empty 1-sorted;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a2;
func FunctionalFAF(A4,A5) -> Function-like quasi_total Relation of the carrier of a3,the carrier of a1 equals
(curry a4) . a5;
end;
:: BILINEAR:def 9
theorem
for b1 being non empty 1-sorted
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b2 holds
FunctionalFAF(b4,b5) = (curry b4) . b5;
:: BILINEAR:funcnot 10 => BILINEAR:func 10
definition
let a1 be non empty 1-sorted;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a3;
func FunctionalSAF(A4,A5) -> Function-like quasi_total Relation of the carrier of a2,the carrier of a1 equals
(curry' a4) . a5;
end;
:: BILINEAR:def 10
theorem
for b1 being non empty 1-sorted
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b3 holds
FunctionalSAF(b4,b5) = (curry' b4) . b5;
:: BILINEAR:th 9
theorem
for b1 being non empty 1-sorted
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b2 holds
dom FunctionalFAF(b4,b5) = the carrier of b3 &
rng FunctionalFAF(b4,b5) c= the carrier of b1 &
(for b6 being Element of the carrier of b3 holds
(FunctionalFAF(b4,b5)) . b6 = b4 .(b5,b6));
:: BILINEAR:th 10
theorem
for b1 being non empty 1-sorted
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b3 holds
dom FunctionalSAF(b4,b5) = the carrier of b2 &
rng FunctionalSAF(b4,b5) c= the carrier of b1 &
(for b6 being Element of the carrier of b2 holds
(FunctionalSAF(b4,b5)) . b6 = b4 .(b6,b5));
:: BILINEAR:th 11
theorem
for b1 being non empty ZeroStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Element of the carrier of b2 holds
FunctionalFAF(NulForm(b2,b3),b4) = 0Functional b3;
:: BILINEAR:th 12
theorem
for b1 being non empty ZeroStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Element of the carrier of b3 holds
FunctionalSAF(NulForm(b2,b3),b4) = 0Functional b2;
:: BILINEAR:th 13
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b6 being Element of the carrier of b3 holds
FunctionalSAF(b4 + b5,b6) = (FunctionalSAF(b4,b6)) + FunctionalSAF(b5,b6);
:: BILINEAR:th 14
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b6 being Element of the carrier of b2 holds
FunctionalFAF(b4 + b5,b6) = (FunctionalFAF(b4,b6)) + FunctionalFAF(b5,b6);
:: BILINEAR:th 15
theorem
for b1 being non empty doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b3 holds
FunctionalSAF(b5 * b4,b6) = b5 * FunctionalSAF(b4,b6);
:: BILINEAR:th 16
theorem
for b1 being non empty doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2 holds
FunctionalFAF(b5 * b4,b6) = b5 * FunctionalFAF(b4,b6);
:: BILINEAR:th 17
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b3 holds
FunctionalSAF(- b4,b5) = - FunctionalSAF(b4,b5);
:: BILINEAR:th 18
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b2 holds
FunctionalFAF(- b4,b5) = - FunctionalFAF(b4,b5);
:: BILINEAR:th 19
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b6 being Element of the carrier of b3 holds
FunctionalSAF(b4 - b5,b6) = (FunctionalSAF(b4,b6)) - FunctionalSAF(b5,b6);
:: BILINEAR:th 20
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b6 being Element of the carrier of b2 holds
FunctionalFAF(b4 - b5,b6) = (FunctionalFAF(b4,b6)) - FunctionalFAF(b5,b6);
:: BILINEAR:funcnot 11 => BILINEAR:func 11
definition
let a1 be non empty multMagma;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
let a5 be Function-like quasi_total Relation of the carrier of a3,the carrier of a1;
func FormFunctional(A4,A5) -> Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1 means
for b1 being Element of the carrier of a2
for b2 being Element of the carrier of a3 holds
it .(b1,b2) = (a4 . b1) * (a5 . b2);
end;
:: BILINEAR:def 11
theorem
for b1 being non empty multMagma
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
for b6 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b6 = FormFunctional(b4,b5)
iff
for b7 being Element of the carrier of b2
for b8 being Element of the carrier of b3 holds
b6 .(b7,b8) = (b4 . b7) * (b5 . b8);
:: BILINEAR:th 21
theorem
for b1 being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b3 holds
(FormFunctional(b4,0Functional b3)) .(b5,b6) = 0. b1;
:: BILINEAR:th 22
theorem
for b1 being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b3 holds
(FormFunctional(0Functional b2,b4)) .(b5,b6) = 0. b1;
:: BILINEAR:th 23
theorem
for b1 being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1 holds
FormFunctional(b4,0Functional b3) = NulForm(b2,b3);
:: BILINEAR:th 24
theorem
for b1 being non empty right_complementable add-associative right_zeroed left-distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1 holds
FormFunctional(0Functional b2,b4) = NulForm(b2,b3);
:: BILINEAR:th 25
theorem
for b1 being non empty multMagma
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
for b6 being Element of the carrier of b2 holds
FunctionalFAF(FormFunctional(b4,b5),b6) = (b4 . b6) * b5;
:: BILINEAR:th 26
theorem
for b1 being non empty commutative multMagma
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
for b6 being Element of the carrier of b3 holds
FunctionalSAF(FormFunctional(b4,b5),b6) = (b5 . b6) * b4;
:: BILINEAR:attrnot 1 => BILINEAR:attr 1
definition
let a1 be non empty addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
attr a4 is additiveFAF means
for b1 being Element of the carrier of a2 holds
FunctionalFAF(a4,b1) is additive(a1, a3);
end;
:: BILINEAR:dfs 11
definiens
let a1 be non empty addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
To prove
a4 is additiveFAF
it is sufficient to prove
thus for b1 being Element of the carrier of a2 holds
FunctionalFAF(a4,b1) is additive(a1, a3);
:: BILINEAR:def 12
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b4 is additiveFAF(b1, b2, b3)
iff
for b5 being Element of the carrier of b2 holds
FunctionalFAF(b4,b5) is additive(b1, b3);
:: BILINEAR:attrnot 2 => BILINEAR:attr 2
definition
let a1 be non empty addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
attr a4 is additiveSAF means
for b1 being Element of the carrier of a3 holds
FunctionalSAF(a4,b1) is additive(a1, a2);
end;
:: BILINEAR:dfs 12
definiens
let a1 be non empty addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
To prove
a4 is additiveSAF
it is sufficient to prove
thus for b1 being Element of the carrier of a3 holds
FunctionalSAF(a4,b1) is additive(a1, a2);
:: BILINEAR:def 13
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b4 is additiveSAF(b1, b2, b3)
iff
for b5 being Element of the carrier of b3 holds
FunctionalSAF(b4,b5) is additive(b1, b2);
:: BILINEAR:attrnot 3 => BILINEAR:attr 3
definition
let a1 be non empty multMagma;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
attr a4 is homogeneousFAF means
for b1 being Element of the carrier of a2 holds
FunctionalFAF(a4,b1) is homogeneous(a1, a3);
end;
:: BILINEAR:dfs 13
definiens
let a1 be non empty multMagma;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
To prove
a4 is homogeneousFAF
it is sufficient to prove
thus for b1 being Element of the carrier of a2 holds
FunctionalFAF(a4,b1) is homogeneous(a1, a3);
:: BILINEAR:def 14
theorem
for b1 being non empty multMagma
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b4 is homogeneousFAF(b1, b2, b3)
iff
for b5 being Element of the carrier of b2 holds
FunctionalFAF(b4,b5) is homogeneous(b1, b3);
:: BILINEAR:attrnot 4 => BILINEAR:attr 4
definition
let a1 be non empty multMagma;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
attr a4 is homogeneousSAF means
for b1 being Element of the carrier of a3 holds
FunctionalSAF(a4,b1) is homogeneous(a1, a2);
end;
:: BILINEAR:dfs 14
definiens
let a1 be non empty multMagma;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
To prove
a4 is homogeneousSAF
it is sufficient to prove
thus for b1 being Element of the carrier of a3 holds
FunctionalSAF(a4,b1) is homogeneous(a1, a2);
:: BILINEAR:def 15
theorem
for b1 being non empty multMagma
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b4 is homogeneousSAF(b1, b2, b3)
iff
for b5 being Element of the carrier of b3 holds
FunctionalSAF(b4,b5) is homogeneous(b1, b2);
:: BILINEAR:funcreg 1
registration
let a1 be non empty right_zeroed addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
cluster NulForm(a2,a3) -> Function-like quasi_total additiveFAF;
end;
:: BILINEAR:funcreg 2
registration
let a1 be non empty right_zeroed addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
cluster NulForm(a2,a3) -> Function-like quasi_total additiveSAF;
end;
:: BILINEAR:exreg 1
registration
let a1 be non empty right_zeroed addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
cluster non empty Relation-like Function-like quasi_total total additiveFAF additiveSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
end;
:: BILINEAR:funcreg 3
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
cluster NulForm(a2,a3) -> Function-like quasi_total homogeneousFAF;
end;
:: BILINEAR:funcreg 4
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
cluster NulForm(a2,a3) -> Function-like quasi_total homogeneousSAF;
end;
:: BILINEAR:exreg 2
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
cluster non empty Relation-like Function-like quasi_total total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
end;
:: BILINEAR:modenot 2
definition
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
mode bilinear-Form of a2,a3 is Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
end;
:: BILINEAR:funcreg 5
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a2;
cluster FunctionalFAF(a4,a5) -> Function-like quasi_total additive;
end;
:: BILINEAR:funcreg 6
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total additiveSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a3;
cluster FunctionalSAF(a4,a5) -> Function-like quasi_total additive;
end;
:: BILINEAR:funcreg 7
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total homogeneousFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a2;
cluster FunctionalFAF(a4,a5) -> Function-like quasi_total homogeneous;
end;
:: BILINEAR:funcreg 8
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a3;
cluster FunctionalSAF(a4,a5) -> Function-like quasi_total homogeneous;
end;
:: BILINEAR:funcreg 9
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
let a5 be Function-like quasi_total additive Relation of the carrier of a3,the carrier of a1;
cluster FormFunctional(a4,a5) -> Function-like quasi_total additiveFAF;
end;
:: BILINEAR:funcreg 10
registration
let a1 be non empty right_complementable add-associative right_zeroed commutative right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total additive Relation of the carrier of a2,the carrier of a1;
let a5 be Function-like quasi_total Relation of the carrier of a3,the carrier of a1;
cluster FormFunctional(a4,a5) -> Function-like quasi_total additiveSAF;
end;
:: BILINEAR:funcreg 11
registration
let a1 be non empty right_complementable add-associative right_zeroed associative commutative right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
let a5 be Function-like quasi_total homogeneous Relation of the carrier of a3,the carrier of a1;
cluster FormFunctional(a4,a5) -> Function-like quasi_total homogeneousFAF;
end;
:: BILINEAR:funcreg 12
registration
let a1 be non empty right_complementable add-associative right_zeroed associative commutative right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total homogeneous Relation of the carrier of a2,the carrier of a1;
let a5 be Function-like quasi_total Relation of the carrier of a3,the carrier of a1;
cluster FormFunctional(a4,a5) -> Function-like quasi_total homogeneousSAF;
end;
:: BILINEAR:funcreg 13
registration
let a1 be non empty non degenerated doubleLoopStr;
let a2 be non empty non trivial VectSpStr over a1;
let a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
let a5 be Function-like quasi_total Relation of the carrier of a3,the carrier of a1;
cluster FormFunctional(a4,a5) -> Function-like quasi_total non trivial;
end;
:: BILINEAR:funcreg 14
registration
let a1 be non empty non degenerated doubleLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be non empty non trivial VectSpStr over a1;
let a4 be Function-like quasi_total Relation of the carrier of a2,the carrier of a1;
let a5 be Function-like quasi_total Relation of the carrier of a3,the carrier of a1;
cluster FormFunctional(a4,a5) -> Function-like quasi_total non trivial;
end;
:: BILINEAR:funcreg 15
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like non constant quasi_total 0-preserving Relation of the carrier of a2,the carrier of a1;
let a5 be Function-like non constant quasi_total 0-preserving Relation of the carrier of a3,the carrier of a1;
cluster FormFunctional(a4,a5) -> Function-like non constant quasi_total;
end;
:: BILINEAR:exreg 3
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
cluster non empty Relation-like Function-like non constant quasi_total total non trivial additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
end;
:: BILINEAR:funcreg 16
registration
let a1 be non empty Abelian add-associative right_zeroed addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total additiveSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster a4 + a5 -> Function-like quasi_total additiveSAF;
end;
:: BILINEAR:funcreg 17
registration
let a1 be non empty Abelian add-associative right_zeroed addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total additiveFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster a4 + a5 -> Function-like quasi_total additiveFAF;
end;
:: BILINEAR:funcreg 18
registration
let a1 be non empty right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total additiveSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a1;
cluster a5 * a4 -> Function-like quasi_total additiveSAF;
end;
:: BILINEAR:funcreg 19
registration
let a1 be non empty right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a1;
cluster a5 * a4 -> Function-like quasi_total additiveFAF;
end;
:: BILINEAR:funcreg 20
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total additiveSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster - a4 -> Function-like quasi_total additiveSAF;
end;
:: BILINEAR:funcreg 21
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster - a4 -> Function-like quasi_total additiveFAF;
end;
:: BILINEAR:funcreg 22
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total additiveSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster a4 - a5 -> Function-like quasi_total additiveSAF;
end;
:: BILINEAR:funcreg 23
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed addLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total additiveFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster a4 - a5 -> Function-like quasi_total additiveFAF;
end;
:: BILINEAR:funcreg 24
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster a4 + a5 -> Function-like quasi_total homogeneousSAF;
end;
:: BILINEAR:funcreg 25
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total homogeneousFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster a4 + a5 -> Function-like quasi_total homogeneousFAF;
end;
:: BILINEAR:funcreg 26
registration
let a1 be non empty right_complementable add-associative right_zeroed associative commutative right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a1;
cluster a5 * a4 -> Function-like quasi_total homogeneousSAF;
end;
:: BILINEAR:funcreg 27
registration
let a1 be non empty right_complementable add-associative right_zeroed associative commutative right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total homogeneousFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
let a5 be Element of the carrier of a1;
cluster a5 * a4 -> Function-like quasi_total homogeneousFAF;
end;
:: BILINEAR:funcreg 28
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster - a4 -> Function-like quasi_total homogeneousSAF;
end;
:: BILINEAR:funcreg 29
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total homogeneousFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster - a4 -> Function-like quasi_total homogeneousFAF;
end;
:: BILINEAR:funcreg 30
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster a4 - a5 -> Function-like quasi_total homogeneousSAF;
end;
:: BILINEAR:funcreg 31
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2, a3 be non empty VectSpStr over a1;
let a4, a5 be Function-like quasi_total homogeneousFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster a4 - a5 -> Function-like quasi_total homogeneousFAF;
end;
:: BILINEAR:th 27
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5 being Element of the carrier of b2
for b6 being Element of the carrier of b3
for b7 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
st b7 is additiveSAF(b1, b2, b3)
holds b7 .(b4 + b5,b6) = (b7 .(b4,b6)) + (b7 .(b5,b6));
:: BILINEAR:th 28
theorem
for b1 being non empty addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Element of the carrier of b2
for b5, b6 being Element of the carrier of b3
for b7 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
st b7 is additiveFAF(b1, b2, b3)
holds b7 .(b4,b5 + b6) = (b7 .(b4,b5)) + (b7 .(b4,b6));
:: BILINEAR:th 29
theorem
for b1 being non empty right_zeroed addLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4, b5 being Element of the carrier of b2
for b6, b7 being Element of the carrier of b3
for b8 being Function-like quasi_total additiveFAF additiveSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b8 .(b4 + b5,b6 + b7) = ((b8 .(b4,b6)) + (b8 .(b4,b7))) + ((b8 .(b5,b6)) + (b8 .(b5,b7)));
:: BILINEAR:th 30
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_zeroed VectSpStr over b1
for b4 being Function-like quasi_total additiveFAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b2 holds
b4 .(b5,0. b3) = 0. b1;
:: BILINEAR:th 31
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_zeroed VectSpStr over b1
for b4 being Function-like quasi_total additiveSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b3 holds
b4 .(0. b2,b5) = 0. b1;
:: BILINEAR:th 32
theorem
for b1 being non empty multMagma
for b2, b3 being non empty VectSpStr over b1
for b4 being Element of the carrier of b2
for b5 being Element of the carrier of b3
for b6 being Element of the carrier of b1
for b7 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
st b7 is homogeneousSAF(b1, b2, b3)
holds b7 .(b6 * b4,b5) = b6 * (b7 .(b4,b5));
:: BILINEAR:th 33
theorem
for b1 being non empty multMagma
for b2, b3 being non empty VectSpStr over b1
for b4 being Element of the carrier of b2
for b5 being Element of the carrier of b3
for b6 being Element of the carrier of b1
for b7 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
st b7 is homogeneousFAF(b1, b2, b3)
holds b7 .(b4,b6 * b5) = b6 * (b7 .(b4,b5));
:: BILINEAR:th 34
theorem
for b1 being non empty right_complementable add-associative right_zeroed associative distributive left_unital doubleLoopStr
for b2, b3 being non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b3 holds
b4 .(0. b2,b5) = 0. b1;
:: BILINEAR:th 35
theorem
for b1 being non empty right_complementable add-associative right_zeroed associative distributive left_unital doubleLoopStr
for b2, b3 being non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total homogeneousFAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Element of the carrier of b2 holds
b4 .(b5,0. b3) = 0. b1;
:: BILINEAR:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4, b5 being Element of the carrier of b2
for b6 being Element of the carrier of b3
for b7 being Function-like quasi_total additiveSAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b7 .(b4 - b5,b6) = (b7 .(b4,b6)) - (b7 .(b5,b6));
:: BILINEAR:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Element of the carrier of b2
for b5, b6 being Element of the carrier of b3
for b7 being Function-like quasi_total additiveFAF homogeneousFAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b7 .(b4,b5 - b6) = (b7 .(b4,b5)) - (b7 .(b4,b6));
:: BILINEAR:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4, b5 being Element of the carrier of b2
for b6, b7 being Element of the carrier of b3
for b8 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b8 .(b4 - b5,b6 - b7) = ((b8 .(b4,b6)) - (b8 .(b4,b7))) - ((b8 .(b5,b6)) - (b8 .(b5,b7)));
:: BILINEAR:th 39
theorem
for b1 being non empty right_complementable add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over b1
for b4, b5 being Element of the carrier of b2
for b6, b7 being Element of the carrier of b3
for b8, b9 being Element of the carrier of b1
for b10 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b10 .(b4 + (b8 * b5),b6 + (b9 * b7)) = ((b10 .(b4,b6)) + (b9 * (b10 .(b4,b7)))) + ((b8 * (b10 .(b5,b6))) + (b8 * (b9 * (b10 .(b5,b7)))));
:: BILINEAR:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4, b5 being Element of the carrier of b2
for b6, b7 being Element of the carrier of b3
for b8, b9 being Element of the carrier of b1
for b10 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b10 .(b4 - (b8 * b5),b6 - (b9 * b7)) = ((b10 .(b4,b6)) - (b9 * (b10 .(b4,b7)))) - ((b8 * (b10 .(b5,b6))) - (b8 * (b9 * (b10 .(b5,b7)))));
:: BILINEAR:th 41
theorem
for b1 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b2, b3 being non empty right_zeroed VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
st (b4 is additiveFAF(b1, b2, b3) or b4 is additiveSAF(b1, b2, b3))
holds b4 is constant
iff
for b5 being Element of the carrier of b2
for b6 being Element of the carrier of b3 holds
b4 .(b5,b6) = 0. b1;
:: BILINEAR:funcnot 12 => BILINEAR:func 12
definition
let a1 be ZeroStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
func leftker A4 -> Element of bool the carrier of a2 equals
{b1 where b1 is Element of the carrier of a2: for b2 being Element of the carrier of a3 holds
a4 .(b1,b2) = 0. a1};
end;
:: BILINEAR:def 16
theorem
for b1 being ZeroStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
leftker b4 = {b5 where b5 is Element of the carrier of b2: for b6 being Element of the carrier of b3 holds
b4 .(b5,b6) = 0. b1};
:: BILINEAR:funcnot 13 => BILINEAR:func 13
definition
let a1 be ZeroStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
func rightker A4 -> Element of bool the carrier of a3 equals
{b1 where b1 is Element of the carrier of a3: for b2 being Element of the carrier of a2 holds
a4 .(b2,b1) = 0. a1};
end;
:: BILINEAR:def 17
theorem
for b1 being ZeroStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
rightker b4 = {b5 where b5 is Element of the carrier of b3: for b6 being Element of the carrier of b2 holds
b4 .(b6,b5) = 0. b1};
:: BILINEAR:funcnot 14 => BILINEAR:func 14
definition
let a1 be ZeroStr;
let a2 be non empty VectSpStr over a1;
let a3 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
func diagker A3 -> Element of bool the carrier of a2 equals
{b1 where b1 is Element of the carrier of a2: a3 .(b1,b1) = 0. a1};
end;
:: BILINEAR:def 18
theorem
for b1 being ZeroStr
for b2 being non empty VectSpStr over b1
for b3 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1 holds
diagker b3 = {b4 where b4 is Element of the carrier of b2: b3 .(b4,b4) = 0. b1};
:: BILINEAR:funcreg 32
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2 be non empty right_zeroed VectSpStr over a1;
let a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total additiveSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster leftker a4 -> non empty;
end;
:: BILINEAR:funcreg 33
registration
let a1 be non empty right_complementable add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over a1;
let a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster leftker a4 -> non empty;
end;
:: BILINEAR:funcreg 34
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be non empty right_zeroed VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster rightker a4 -> non empty;
end;
:: BILINEAR:funcreg 35
registration
let a1 be non empty right_complementable add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total homogeneousFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster rightker a4 -> non empty;
end;
:: BILINEAR:funcreg 36
registration
let a1 be non empty right_complementable add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_zeroed VectSpStr over a1;
let a3 be Function-like quasi_total additiveFAF Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
cluster diagker a3 -> non empty;
end;
:: BILINEAR:funcreg 37
registration
let a1 be non empty right_complementable add-associative right_zeroed doubleLoopStr;
let a2 be non empty right_zeroed VectSpStr over a1;
let a3 be Function-like quasi_total additiveSAF Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
cluster diagker a3 -> non empty;
end;
:: BILINEAR:funcreg 38
registration
let a1 be non empty right_complementable add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over a1;
let a3 be Function-like quasi_total homogeneousFAF Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
cluster diagker a3 -> non empty;
end;
:: BILINEAR:funcreg 39
registration
let a1 be non empty right_complementable add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable add-associative right_zeroed VectSp-like VectSpStr over a1;
let a3 be Function-like quasi_total homogeneousSAF Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
cluster diagker a3 -> non empty;
end;
:: BILINEAR:th 42
theorem
for b1 being ZeroStr
for b2 being non empty VectSpStr over b1
for b3 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1 holds
leftker b3 c= diagker b3 & rightker b3 c= diagker b3;
:: BILINEAR:th 43
theorem
for b1 being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total additiveSAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
leftker b4 is linearly-closed(b1, b2);
:: BILINEAR:th 44
theorem
for b1 being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total additiveFAF homogeneousFAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
rightker b4 is linearly-closed(b1, b3);
:: BILINEAR:funcnot 15 => BILINEAR:func 15
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total additiveSAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
func LKer A4 -> non empty strict Subspace of a2 means
the carrier of it = leftker a4;
end;
:: BILINEAR:def 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total additiveSAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being non empty strict Subspace of b2 holds
b5 = LKer b4
iff
the carrier of b5 = leftker b4;
:: BILINEAR:funcnot 16 => BILINEAR:func 16
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF homogeneousFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
func RKer A4 -> non empty strict Subspace of a3 means
the carrier of it = rightker a4;
end;
:: BILINEAR:def 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty VectSpStr over b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total additiveFAF homogeneousFAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being non empty strict Subspace of b3 holds
b5 = RKer b4
iff
the carrier of b5 = rightker b4;
:: BILINEAR:funcnot 17 => BILINEAR:func 17
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total additiveSAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
func LQForm A4 -> Function-like quasi_total additiveSAF homogeneousSAF Relation of [:the carrier of VectQuot(a2,LKer a4),the carrier of a3:],the carrier of a1 means
for b1 being Element of the carrier of VectQuot(a2,LKer a4)
for b2 being Element of the carrier of a3
for b3 being Element of the carrier of a2
st b1 = b3 + LKer a4
holds it .(b1,b2) = a4 .(b3,b2);
end;
:: BILINEAR:def 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total additiveSAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Function-like quasi_total additiveSAF homogeneousSAF Relation of [:the carrier of VectQuot(b2,LKer b4),the carrier of b3:],the carrier of b1 holds
b5 = LQForm b4
iff
for b6 being Element of the carrier of VectQuot(b2,LKer b4)
for b7 being Element of the carrier of b3
for b8 being Element of the carrier of b2
st b6 = b8 + LKer b4
holds b5 .(b6,b7) = b4 .(b8,b7);
:: BILINEAR:funcnot 18 => BILINEAR:func 18
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF homogeneousFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
func RQForm A4 -> Function-like quasi_total additiveFAF homogeneousFAF Relation of [:the carrier of a2,the carrier of VectQuot(a3,RKer a4):],the carrier of a1 means
for b1 being Element of the carrier of VectQuot(a3,RKer a4)
for b2 being Element of the carrier of a2
for b3 being Element of the carrier of a3
st b1 = b3 + RKer a4
holds it .(b2,b1) = a4 .(b2,b3);
end;
:: BILINEAR:def 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty VectSpStr over b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total additiveFAF homogeneousFAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Function-like quasi_total additiveFAF homogeneousFAF Relation of [:the carrier of b2,the carrier of VectQuot(b3,RKer b4):],the carrier of b1 holds
b5 = RQForm b4
iff
for b6 being Element of the carrier of VectQuot(b3,RKer b4)
for b7 being Element of the carrier of b2
for b8 being Element of the carrier of b3
st b6 = b8 + RKer b4
holds b5 .(b7,b6) = b4 .(b7,b8);
:: BILINEAR:funcreg 40
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster LQForm a4 -> Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF;
end;
:: BILINEAR:funcreg 41
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster RQForm a4 -> Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF;
end;
:: BILINEAR:funcnot 19 => BILINEAR:func 19
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
func QForm A4 -> Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of VectQuot(a2,LKer a4),the carrier of VectQuot(a3,RKer a4):],the carrier of a1 means
for b1 being Element of the carrier of VectQuot(a2,LKer a4)
for b2 being Element of the carrier of VectQuot(a3,RKer a4)
for b3 being Element of the carrier of a2
for b4 being Element of the carrier of a3
st b1 = b3 + LKer a4 & b2 = b4 + RKer a4
holds it .(b1,b2) = a4 .(b3,b4);
end;
:: BILINEAR:def 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1
for b5 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of VectQuot(b2,LKer b4),the carrier of VectQuot(b3,RKer b4):],the carrier of b1 holds
b5 = QForm b4
iff
for b6 being Element of the carrier of VectQuot(b2,LKer b4)
for b7 being Element of the carrier of VectQuot(b3,RKer b4)
for b8 being Element of the carrier of b2
for b9 being Element of the carrier of b3
st b6 = b8 + LKer b4 & b7 = b9 + RKer b4
holds b5 .(b6,b7) = b4 .(b8,b9);
:: BILINEAR:th 45
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total additiveSAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
rightker b4 = rightker LQForm b4;
:: BILINEAR:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2 being non empty VectSpStr over b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total additiveFAF homogeneousFAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
leftker b4 = leftker RQForm b4;
:: BILINEAR:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
RKer b4 = RKer LQForm b4;
:: BILINEAR:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
LKer b4 = LKer RQForm b4;
:: BILINEAR:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
QForm b4 = RQForm LQForm b4 & QForm b4 = LQForm RQForm b4;
:: BILINEAR:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
leftker QForm b4 = leftker RQForm LQForm b4 &
rightker QForm b4 = rightker RQForm LQForm b4 &
leftker QForm b4 = leftker LQForm RQForm b4 &
rightker QForm b4 = rightker LQForm RQForm b4;
:: BILINEAR:th 51
theorem
for b1 being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1 holds
ker b4 c= leftker FormFunctional(b4,b5);
:: BILINEAR:th 52
theorem
for b1 being non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
st b5 <> 0Functional b3
holds leftker FormFunctional(b4,b5) = ker b4;
:: BILINEAR:th 53
theorem
for b1 being non empty right_complementable add-associative right_zeroed distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1 holds
ker b5 c= rightker FormFunctional(b4,b5);
:: BILINEAR:th 54
theorem
for b1 being non empty right_complementable almost_left_invertible add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
st b4 <> 0Functional b2
holds rightker FormFunctional(b4,b5) = ker b5;
:: BILINEAR:th 55
theorem
for b1 being non empty right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total additive homogeneous Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
st b5 <> 0Functional b3
holds LKer FormFunctional(b4,b5) = Ker b4 &
LQForm FormFunctional(b4,b5) = FormFunctional(CQFunctional b4,b5);
:: BILINEAR:th 56
theorem
for b1 being non empty right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2 being non empty VectSpStr over b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like quasi_total Relation of the carrier of b2,the carrier of b1
for b5 being Function-like quasi_total additive homogeneous Relation of the carrier of b3,the carrier of b1
st b4 <> 0Functional b2
holds RKer FormFunctional(b4,b5) = Ker b5 &
RQForm FormFunctional(b4,b5) = FormFunctional(b4,CQFunctional b5);
:: BILINEAR:th 57
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr
for b2, b3 being non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over b1
for b4 being Function-like non constant quasi_total additive homogeneous Relation of the carrier of b2,the carrier of b1
for b5 being Function-like non constant quasi_total additive homogeneous Relation of the carrier of b3,the carrier of b1 holds
QForm FormFunctional(b4,b5) = FormFunctional(CQFunctional b4,CQFunctional b5);
:: BILINEAR:attrnot 5 => BILINEAR:attr 5
definition
let a1 be ZeroStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
attr a4 is degenerated-on-left means
leftker a4 <> {0. a2};
end;
:: BILINEAR:dfs 23
definiens
let a1 be ZeroStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
To prove
a4 is degenerated-on-left
it is sufficient to prove
thus leftker a4 <> {0. a2};
:: BILINEAR:def 24
theorem
for b1 being ZeroStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b4 is degenerated-on-left(b1, b2, b3)
iff
leftker b4 <> {0. b2};
:: BILINEAR:attrnot 6 => BILINEAR:attr 6
definition
let a1 be ZeroStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
attr a4 is degenerated-on-right means
rightker a4 <> {0. a3};
end;
:: BILINEAR:dfs 24
definiens
let a1 be ZeroStr;
let a2, a3 be non empty VectSpStr over a1;
let a4 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
To prove
a4 is degenerated-on-right
it is sufficient to prove
thus rightker a4 <> {0. a3};
:: BILINEAR:def 25
theorem
for b1 being ZeroStr
for b2, b3 being non empty VectSpStr over b1
for b4 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b3:],the carrier of b1 holds
b4 is degenerated-on-right(b1, b2, b3)
iff
rightker b4 <> {0. b3};
:: BILINEAR:funcreg 42
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a3 be non empty right_zeroed VectSpStr over a1;
let a4 be Function-like quasi_total additiveSAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster LQForm a4 -> Function-like quasi_total additiveSAF homogeneousSAF non degenerated-on-left;
end;
:: BILINEAR:funcreg 43
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2 be non empty right_zeroed VectSpStr over a1;
let a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF homogeneousFAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster RQForm a4 -> Function-like quasi_total additiveFAF homogeneousFAF non degenerated-on-right;
end;
:: BILINEAR:funcreg 44
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster QForm a4 -> Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF non degenerated-on-left non degenerated-on-right;
end;
:: BILINEAR:funcreg 45
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster RQForm LQForm a4 -> Function-like quasi_total additiveFAF homogeneousFAF non degenerated-on-left non degenerated-on-right;
end;
:: BILINEAR:funcreg 46
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive doubleLoopStr;
let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster LQForm RQForm a4 -> Function-like quasi_total additiveSAF homogeneousSAF non degenerated-on-left non degenerated-on-right;
end;
:: BILINEAR:funcreg 47
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2, a3 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
let a4 be Function-like non constant quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF Relation of [:the carrier of a2,the carrier of a3:],the carrier of a1;
cluster QForm a4 -> Function-like non constant quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF;
end;
:: BILINEAR:attrnot 7 => BILINEAR:attr 7
definition
let a1 be 1-sorted;
let a2 be VectSpStr over a1;
let a3 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
attr a3 is symmetric means
for b1, b2 being Element of the carrier of a2 holds
a3 .(b1,b2) = a3 .(b2,b1);
end;
:: BILINEAR:dfs 25
definiens
let a1 be 1-sorted;
let a2 be VectSpStr over a1;
let a3 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
To prove
a3 is symmetric
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a2 holds
a3 .(b1,b2) = a3 .(b2,b1);
:: BILINEAR:def 26
theorem
for b1 being 1-sorted
for b2 being VectSpStr over b1
for b3 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1 holds
b3 is symmetric(b1, b2)
iff
for b4, b5 being Element of the carrier of b2 holds
b3 .(b4,b5) = b3 .(b5,b4);
:: BILINEAR:attrnot 8 => BILINEAR:attr 8
definition
let a1 be ZeroStr;
let a2 be VectSpStr over a1;
let a3 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
attr a3 is alternating means
for b1 being Element of the carrier of a2 holds
a3 .(b1,b1) = 0. a1;
end;
:: BILINEAR:dfs 26
definiens
let a1 be ZeroStr;
let a2 be VectSpStr over a1;
let a3 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
To prove
a3 is alternating
it is sufficient to prove
thus for b1 being Element of the carrier of a2 holds
a3 .(b1,b1) = 0. a1;
:: BILINEAR:def 27
theorem
for b1 being ZeroStr
for b2 being VectSpStr over b1
for b3 being Function-like quasi_total Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1 holds
b3 is alternating(b1, b2)
iff
for b4 being Element of the carrier of b2 holds
b3 .(b4,b4) = 0. b1;
:: BILINEAR:funcreg 48
registration
let a1 be non empty ZeroStr;
let a2 be non empty VectSpStr over a1;
cluster NulForm(a2,a2) -> Function-like quasi_total symmetric;
end;
:: BILINEAR:funcreg 49
registration
let a1 be non empty ZeroStr;
let a2 be non empty VectSpStr over a1;
cluster NulForm(a2,a2) -> Function-like quasi_total alternating;
end;
:: BILINEAR:exreg 4
registration
let a1 be non empty ZeroStr;
let a2 be non empty VectSpStr over a1;
cluster non empty Relation-like Function-like quasi_total total symmetric Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
end;
:: BILINEAR:exreg 5
registration
let a1 be non empty ZeroStr;
let a2 be non empty VectSpStr over a1;
cluster non empty Relation-like Function-like quasi_total total alternating Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
end;
:: BILINEAR:exreg 6
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2 be non empty VectSpStr over a1;
cluster non empty Relation-like Function-like quasi_total total additiveFAF additiveSAF homogeneousFAF homogeneousSAF symmetric Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
end;
:: BILINEAR:exreg 7
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2 be non empty VectSpStr over a1;
cluster non empty Relation-like Function-like quasi_total total additiveFAF additiveSAF homogeneousFAF homogeneousSAF alternating Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
end;
:: BILINEAR:exreg 8
registration
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive doubleLoopStr;
let a2 be non empty non trivial right_complementable Abelian add-associative right_zeroed VectSp-like VectSpStr over a1;
cluster non empty Relation-like Function-like non constant quasi_total total non trivial additiveFAF additiveSAF homogeneousFAF homogeneousSAF symmetric Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
end;
:: BILINEAR:exreg 9
registration
let a1 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a2 be non empty VectSpStr over a1;
cluster non empty Relation-like Function-like quasi_total total additiveFAF additiveSAF alternating Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
end;
:: BILINEAR:funcreg 50
registration
let a1 be non empty addLoopStr;
let a2 be non empty VectSpStr over a1;
let a3, a4 be Function-like quasi_total symmetric Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
cluster a3 + a4 -> Function-like quasi_total symmetric;
end;
:: BILINEAR:funcreg 51
registration
let a1 be non empty doubleLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be Function-like quasi_total symmetric Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
let a4 be Element of the carrier of a1;
cluster a4 * a3 -> Function-like quasi_total symmetric;
end;
:: BILINEAR:funcreg 52
registration
let a1 be non empty addLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be Function-like quasi_total symmetric Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
cluster - a3 -> Function-like quasi_total symmetric;
end;
:: BILINEAR:funcreg 53
registration
let a1 be non empty addLoopStr;
let a2 be non empty VectSpStr over a1;
let a3, a4 be Function-like quasi_total symmetric Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
cluster a3 - a4 -> Function-like quasi_total symmetric;
end;
:: BILINEAR:funcreg 54
registration
let a1 be non empty right_zeroed addLoopStr;
let a2 be non empty VectSpStr over a1;
let a3, a4 be Function-like quasi_total alternating Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
cluster a3 + a4 -> Function-like quasi_total alternating;
end;
:: BILINEAR:funcreg 55
registration
let a1 be non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be Function-like quasi_total alternating Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
let a4 be Element of the carrier of a1;
cluster a4 * a3 -> Function-like quasi_total alternating;
end;
:: BILINEAR:funcreg 56
registration
let a1 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be Function-like quasi_total alternating Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
cluster - a3 -> Function-like quasi_total alternating;
end;
:: BILINEAR:funcreg 57
registration
let a1 be non empty right_complementable add-associative right_zeroed addLoopStr;
let a2 be non empty VectSpStr over a1;
let a3, a4 be Function-like quasi_total alternating Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
cluster a3 - a4 -> Function-like quasi_total alternating;
end;
:: BILINEAR:th 58
theorem
for b1 being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for b2 being non empty VectSpStr over b1
for b3 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF symmetric Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1 holds
leftker b3 = rightker b3;
:: BILINEAR:th 59
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being non empty VectSpStr over b1
for b3 being Function-like quasi_total additiveFAF additiveSAF alternating Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1
for b4, b5 being Element of the carrier of b2 holds
b3 .(b4,b5) = - (b3 .(b5,b4));
:: BILINEAR:attrnot 9 => BILINEAR:attr 8
definition
let a1 be ZeroStr;
let a2 be VectSpStr over a1;
let a3 be Function-like quasi_total Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
attr a3 is alternating means
for b1, b2 being Element of the carrier of a2 holds
a3 .(b1,b2) = - (a3 .(b2,b1));
end;
:: BILINEAR:dfs 27
definiens
let a1 be non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive Fanoian doubleLoopStr;
let a2 be non empty VectSpStr over a1;
let a3 be Function-like quasi_total additiveFAF additiveSAF Relation of [:the carrier of a2,the carrier of a2:],the carrier of a1;
To prove
a3 is alternating
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a2 holds
a3 .(b1,b2) = - (a3 .(b2,b1));
:: BILINEAR:def 28
theorem
for b1 being non empty non degenerated right_complementable almost_left_invertible Abelian add-associative right_zeroed associative commutative well-unital distributive Fanoian doubleLoopStr
for b2 being non empty VectSpStr over b1
for b3 being Function-like quasi_total additiveFAF additiveSAF Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1 holds
b3 is alternating(b1, b2)
iff
for b4, b5 being Element of the carrier of b2 holds
b3 .(b4,b5) = - (b3 .(b5,b4));
:: BILINEAR:th 60
theorem
for b1 being non empty right_complementable add-associative right_zeroed right-distributive doubleLoopStr
for b2 being non empty VectSpStr over b1
for b3 being Function-like quasi_total additiveFAF additiveSAF homogeneousFAF homogeneousSAF alternating Relation of [:the carrier of b2,the carrier of b2:],the carrier of b1 holds
leftker b3 = rightker b3;