Article ANALORT, MML version 4.99.1005
:: ANALORT:funcnot 1 => ANALORT:func 1
definition
let a1 be non empty Abelian addLoopStr;
let a2, a3 be Element of the carrier of a1;
redefine func a2 + a3 -> Element of the carrier of a1;
commutativity;
:: for a1 being non empty Abelian addLoopStr
:: for a2, a3 being Element of the carrier of a1 holds
:: a2 + a3 = a3 + a2;
end;
:: ANALORT:funcnot 2 => ANALORT:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4 be Element of the carrier of a1;
func Ortm(A2,A3,A4) -> Element of the carrier of a1 equals
((pr1(a2,a3,a4)) * a2) + ((- pr2(a2,a3,a4)) * a3);
end;
:: ANALORT:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1 holds
Ortm(b2,b3,b4) = ((pr1(b2,b3,b4)) * b2) + ((- pr2(b2,b3,b4)) * b3);
:: ANALORT:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st Gen b2,b3
holds Ortm(b2,b3,b4 + b5) = (Ortm(b2,b3,b4)) + Ortm(b2,b3,b5);
:: ANALORT:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of REAL
st Gen b2,b3
holds Ortm(b2,b3,b5 * b4) = b5 * Ortm(b2,b3,b4);
:: ANALORT:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st Gen b2,b3
holds Ortm(b2,b3,0. b1) = 0. b1;
:: ANALORT:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st Gen b2,b3
holds Ortm(b2,b3,- b4) = - Ortm(b2,b3,b4);
:: ANALORT:th 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st Gen b2,b3
holds Ortm(b2,b3,b4 - b5) = (Ortm(b2,b3,b4)) - Ortm(b2,b3,b5);
:: ANALORT:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st Gen b2,b3 & Ortm(b2,b3,b4) = Ortm(b2,b3,b5)
holds b4 = b5;
:: ANALORT:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st Gen b2,b3
holds Ortm(b2,b3,Ortm(b2,b3,b4)) = b4;
:: ANALORT:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st Gen b2,b3
holds ex b5 being Element of the carrier of b1 st
b4 = Ortm(b2,b3,b5);
:: ANALORT:funcnot 3 => ANALORT:func 3
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4 be Element of the carrier of a1;
func Orte(A2,A3,A4) -> Element of the carrier of a1 equals
((pr2(a2,a3,a4)) * a2) + ((- pr1(a2,a3,a4)) * a3);
end;
:: ANALORT:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1 holds
Orte(b2,b3,b4) = ((pr2(b2,b3,b4)) * b2) + ((- pr1(b2,b3,b4)) * b3);
:: ANALORT:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st Gen b2,b3
holds Orte(b2,b3,- b4) = - Orte(b2,b3,b4);
:: ANALORT:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st Gen b2,b3
holds Orte(b2,b3,b4 + b5) = (Orte(b2,b3,b4)) + Orte(b2,b3,b5);
:: ANALORT:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st Gen b2,b3
holds Orte(b2,b3,b4 - b5) = (Orte(b2,b3,b4)) - Orte(b2,b3,b5);
:: ANALORT:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of REAL
st Gen b2,b3
holds Orte(b2,b3,b5 * b4) = b5 * Orte(b2,b3,b4);
:: ANALORT:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st Gen b2,b3 & Orte(b2,b3,b4) = Orte(b2,b3,b5)
holds b4 = b5;
:: ANALORT:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st Gen b2,b3
holds Orte(b2,b3,Orte(b2,b3,b4)) = - b4;
:: ANALORT:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Element of the carrier of b1
st Gen b2,b3
holds ex b5 being Element of the carrier of b1 st
Orte(b2,b3,b5) = b4;
:: ANALORT:prednot 1 => ANALORT:pred 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7 be Element of the carrier of a1;
pred A4,A5,A6,A7 are_COrte_wrt A2,A3 means
Orte(a2,a3,a4),Orte(a2,a3,a5) // a6,a7;
end;
:: ANALORT:dfs 3
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7 be Element of the carrier of a1;
To prove
a4,a5,a6,a7 are_COrte_wrt a2,a3
it is sufficient to prove
thus Orte(a2,a3,a4),Orte(a2,a3,a5) // a6,a7;
:: ANALORT:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
b4,b5,b6,b7 are_COrte_wrt b2,b3
iff
Orte(b2,b3,b4),Orte(b2,b3,b5) // b6,b7;
:: ANALORT:prednot 2 => ANALORT:pred 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7 be Element of the carrier of a1;
pred A4,A5,A6,A7 are_COrtm_wrt A2,A3 means
Ortm(a2,a3,a4),Ortm(a2,a3,a5) // a6,a7;
end;
:: ANALORT:dfs 4
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5, a6, a7 be Element of the carrier of a1;
To prove
a4,a5,a6,a7 are_COrtm_wrt a2,a3
it is sufficient to prove
thus Ortm(a2,a3,a4),Ortm(a2,a3,a5) // a6,a7;
:: ANALORT:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1 holds
b4,b5,b6,b7 are_COrtm_wrt b2,b3
iff
Ortm(b2,b3,b4),Ortm(b2,b3,b5) // b6,b7;
:: ANALORT:th 16
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st Gen b2,b3 & b4,b5 // b6,b7
holds Orte(b2,b3,b4),Orte(b2,b3,b5) // Orte(b2,b3,b6),Orte(b2,b3,b7);
:: ANALORT:th 17
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st Gen b2,b3 & b4,b5 // b6,b7
holds Ortm(b2,b3,b4),Ortm(b2,b3,b5) // Ortm(b2,b3,b6),Ortm(b2,b3,b7);
:: ANALORT:th 18
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3
holds b6,b7,b5,b4 are_COrte_wrt b2,b3;
:: ANALORT:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3
holds b6,b7,b4,b5 are_COrtm_wrt b2,b3;
:: ANALORT:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
b2,b2,b3,b4 are_COrte_wrt b5,b6;
:: ANALORT:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
b2,b2,b3,b4 are_COrtm_wrt b5,b6;
:: ANALORT:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
b2,b3,b4,b4 are_COrte_wrt b5,b6;
:: ANALORT:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6 being Element of the carrier of b1 holds
b2,b3,b4,b4 are_COrtm_wrt b5,b6;
:: ANALORT:th 24
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1
st Gen b2,b3
holds b4,b5,Orte(b2,b3,b4),Orte(b2,b3,b5) are_Ort_wrt b2,b3;
:: ANALORT:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2,b3,Orte(b4,b5,b2),Orte(b4,b5,b3) are_COrte_wrt b4,b5;
:: ANALORT:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2,b3,Ortm(b4,b5,b2),Ortm(b4,b5,b3) are_COrtm_wrt b4,b5;
:: ANALORT:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st Gen b2,b3
holds b4,b5 // b6,b7
iff
ex b8, b9 being Element of the carrier of b1 st
b8 <> b9 & b8,b9,b4,b5 are_COrte_wrt b2,b3 & b8,b9,b6,b7 are_COrte_wrt b2,b3;
:: ANALORT:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st Gen b2,b3
holds b4,b5 // b6,b7
iff
ex b8, b9 being Element of the carrier of b1 st
b8 <> b9 & b8,b9,b4,b5 are_COrtm_wrt b2,b3 & b8,b9,b6,b7 are_COrtm_wrt b2,b3;
:: ANALORT:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st Gen b2,b3
holds b4,b5,b6,b7 are_Ort_wrt b2,b3
iff
(b4,b5,b6,b7 are_COrte_wrt b2,b3 or b4,b5,b7,b6 are_COrte_wrt b2,b3);
:: ANALORT:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b4,b5,b7,b6 are_COrte_wrt b2,b3 & b4 <> b5
holds b6 = b7;
:: ANALORT:th 31
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b4,b5,b7,b6 are_COrtm_wrt b2,b3 & b4 <> b5
holds b6 = b7;
:: ANALORT:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b4,b5,b6,b8 are_COrte_wrt b2,b3 & not b4,b5,b7,b8 are_COrte_wrt b2,b3
holds b4,b5,b8,b7 are_COrte_wrt b2,b3;
:: ANALORT:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b4,b5,b6,b8 are_COrtm_wrt b2,b3 & not b4,b5,b7,b8 are_COrtm_wrt b2,b3
holds b4,b5,b8,b7 are_COrtm_wrt b2,b3;
:: ANALORT:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st b2,b3,b4,b5 are_COrte_wrt b6,b7
holds b3,b2,b5,b4 are_COrte_wrt b6,b7;
:: ANALORT:th 35
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st b2,b3,b4,b5 are_COrtm_wrt b6,b7
holds b3,b2,b5,b4 are_COrtm_wrt b6,b7;
:: ANALORT:th 36
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b4,b5,b7,b8 are_COrte_wrt b2,b3
holds b4,b5,b6,b8 are_COrte_wrt b2,b3;
:: ANALORT:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b4,b5,b7,b8 are_COrtm_wrt b2,b3
holds b4,b5,b6,b8 are_COrtm_wrt b2,b3;
:: ANALORT:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st Gen b2,b3
for b4, b5, b6 being Element of the carrier of b1 holds
ex b7 being Element of the carrier of b1 st
b6 <> b7 & b6,b7,b4,b5 are_COrte_wrt b2,b3;
:: ANALORT:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st Gen b2,b3
for b4, b5, b6 being Element of the carrier of b1 holds
ex b7 being Element of the carrier of b1 st
b6 <> b7 & b6,b7,b4,b5 are_COrtm_wrt b2,b3;
:: ANALORT:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st Gen b2,b3
for b4, b5, b6 being Element of the carrier of b1 holds
ex b7 being Element of the carrier of b1 st
b6 <> b7 & b4,b5,b6,b7 are_COrte_wrt b2,b3;
:: ANALORT:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st Gen b2,b3
for b4, b5, b6 being Element of the carrier of b1 holds
ex b7 being Element of the carrier of b1 st
b6 <> b7 & b4,b5,b6,b7 are_COrtm_wrt b2,b3;
:: ANALORT:th 42
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b8,b9,b6,b7 are_COrte_wrt b2,b3 & b8,b9,b10,b11 are_COrte_wrt b2,b3 & b8 <> b9 & b6 <> b7
holds b4,b5,b10,b11 are_COrte_wrt b2,b3;
:: ANALORT:th 43
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b8,b9,b6,b7 are_COrtm_wrt b2,b3 & b8,b9,b10,b11 are_COrtm_wrt b2,b3 & b8 <> b9 & b6 <> b7
holds b4,b5,b10,b11 are_COrtm_wrt b2,b3;
:: ANALORT:th 46
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b6,b7,b8,b9 are_COrte_wrt b2,b3 & b10,b11,b8,b9 are_COrte_wrt b2,b3 & not b4,b5,b10,b11 are_COrte_wrt b2,b3 & b6 <> b7
holds b8 = b9;
:: ANALORT:th 47
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b6,b7,b8,b9 are_COrtm_wrt b2,b3 & b10,b11,b8,b9 are_COrtm_wrt b2,b3 & not b4,b5,b10,b11 are_COrtm_wrt b2,b3 & b6 <> b7
holds b8 = b9;
:: ANALORT:th 48
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrte_wrt b2,b3 & b6,b7,b8,b9 are_COrte_wrt b2,b3 & b4,b5,b10,b11 are_COrte_wrt b2,b3 & not b10,b11,b8,b9 are_COrte_wrt b2,b3 & b6 <> b7
holds b4 = b5;
:: ANALORT:th 49
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7, b8, b9, b10, b11 being Element of the carrier of b1
st Gen b2,b3 & b4,b5,b6,b7 are_COrtm_wrt b2,b3 & b6,b7,b8,b9 are_COrtm_wrt b2,b3 & b4,b5,b10,b11 are_COrtm_wrt b2,b3 & not b10,b11,b8,b9 are_COrtm_wrt b2,b3 & b6 <> b7
holds b4 = b5;
:: ANALORT:th 50
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st Gen b2,b3
for b4, b5, b6, b7, b8 being Element of the carrier of b1
st not b4,b7,b5,b6 are_COrte_wrt b2,b3 & not b4,b7,b6,b5 are_COrte_wrt b2,b3 & b6,b8,b6,b5 are_COrte_wrt b2,b3
holds ex b9 being Element of the carrier of b1 st
(b4,b7,b4,b9 are_COrte_wrt b2,b3 or b4,b7,b9,b4 are_COrte_wrt b2,b3) & (b6,b8,b6,b9 are_COrte_wrt b2,b3 or b6,b8,b9,b6 are_COrte_wrt b2,b3);
:: ANALORT:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st Gen b2,b3
holds ex b4, b5, b6 being Element of the carrier of b1 st
b4,b5,b4,b6 are_COrte_wrt b2,b3 &
(for b7, b8 being Element of the carrier of b1
st b7,b8,b4,b5 are_COrte_wrt b2,b3 & (b7,b8,b4,b6 are_COrte_wrt b2,b3 or b7,b8,b6,b4 are_COrte_wrt b2,b3)
holds b7 = b8);
:: ANALORT:th 52
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st Gen b2,b3
for b4, b5, b6, b7, b8 being Element of the carrier of b1
st not b4,b7,b5,b6 are_COrtm_wrt b2,b3 & not b4,b7,b6,b5 are_COrtm_wrt b2,b3 & b6,b8,b6,b5 are_COrtm_wrt b2,b3
holds ex b9 being Element of the carrier of b1 st
(b4,b7,b4,b9 are_COrtm_wrt b2,b3 or b4,b7,b9,b4 are_COrtm_wrt b2,b3) & (b6,b8,b6,b9 are_COrtm_wrt b2,b3 or b6,b8,b9,b6 are_COrtm_wrt b2,b3);
:: ANALORT:th 53
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st Gen b2,b3
holds ex b4, b5, b6 being Element of the carrier of b1 st
b4,b5,b4,b6 are_COrtm_wrt b2,b3 &
(for b7, b8 being Element of the carrier of b1
st b7,b8,b4,b5 are_COrtm_wrt b2,b3 & (b7,b8,b4,b6 are_COrtm_wrt b2,b3 or b7,b8,b6,b4 are_COrtm_wrt b2,b3)
holds b7 = b8);
:: ANALORT:funcnot 4 => ANALORT:func 4
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3 be Element of the carrier of a1;
func CORTE(A1,A2,A3) -> Relation of [:the carrier of a1,the carrier of a1:],[:the carrier of a1,the carrier of a1:] means
for b1, b2 being set holds
[b1,b2] in it
iff
ex b3, b4, b5, b6 being Element of the carrier of a1 st
b1 = [b3,b4] & b2 = [b5,b6] & b3,b4,b5,b6 are_COrte_wrt a2,a3;
end;
:: ANALORT:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Relation of [:the carrier of b1,the carrier of b1:],[:the carrier of b1,the carrier of b1:] holds
b4 = CORTE(b1,b2,b3)
iff
for b5, b6 being set holds
[b5,b6] in b4
iff
ex b7, b8, b9, b10 being Element of the carrier of b1 st
b5 = [b7,b8] & b6 = [b9,b10] & b7,b8,b9,b10 are_COrte_wrt b2,b3;
:: ANALORT:funcnot 5 => ANALORT:func 5
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3 be Element of the carrier of a1;
func CORTM(A1,A2,A3) -> Relation of [:the carrier of a1,the carrier of a1:],[:the carrier of a1,the carrier of a1:] means
for b1, b2 being set holds
[b1,b2] in it
iff
ex b3, b4, b5, b6 being Element of the carrier of a1 st
b1 = [b3,b4] & b2 = [b5,b6] & b3,b4,b5,b6 are_COrtm_wrt a2,a3;
end;
:: ANALORT:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being Relation of [:the carrier of b1,the carrier of b1:],[:the carrier of b1,the carrier of b1:] holds
b4 = CORTM(b1,b2,b3)
iff
for b5, b6 being set holds
[b5,b6] in b4
iff
ex b7, b8, b9, b10 being Element of the carrier of b1 st
b5 = [b7,b8] & b6 = [b9,b10] & b7,b8,b9,b10 are_COrtm_wrt b2,b3;
:: ANALORT:funcnot 6 => ANALORT:func 6
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3 be Element of the carrier of a1;
func CESpace(A1,A2,A3) -> strict AffinStruct equals
AffinStruct(#the carrier of a1,CORTE(a1,a2,a3)#);
end;
:: ANALORT:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
CESpace(b1,b2,b3) = AffinStruct(#the carrier of b1,CORTE(b1,b2,b3)#);
:: ANALORT:funcreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3 be Element of the carrier of a1;
cluster CESpace(a1,a2,a3) -> non empty strict;
end;
:: ANALORT:funcnot 7 => ANALORT:func 7
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3 be Element of the carrier of a1;
func CMSpace(A1,A2,A3) -> strict AffinStruct equals
AffinStruct(#the carrier of a1,CORTM(a1,a2,a3)#);
end;
:: ANALORT:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
CMSpace(b1,b2,b3) = AffinStruct(#the carrier of b1,CORTM(b1,b2,b3)#);
:: ANALORT:funcreg 2
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3 be Element of the carrier of a1;
cluster CMSpace(a1,a2,a3) -> non empty strict;
end;
:: ANALORT:th 54
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being set holds
b4 is Element of the carrier of CESpace(b1,b2,b3)
iff
b4 is Element of the carrier of b1;
:: ANALORT:th 55
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
for b4 being set holds
b4 is Element of the carrier of CMSpace(b1,b2,b3)
iff
b4 is Element of the carrier of b1;
:: ANALORT:th 56
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
for b8, b9, b10, b11 being Element of the carrier of CESpace(b1,b6,b7)
st b2 = b8 & b3 = b9 & b4 = b10 & b5 = b11
holds b8,b9 // b10,b11
iff
b2,b3,b4,b5 are_COrte_wrt b6,b7;
:: ANALORT:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
for b8, b9, b10, b11 being Element of the carrier of CMSpace(b1,b6,b7)
st b2 = b8 & b3 = b9 & b4 = b10 & b5 = b11
holds b8,b9 // b10,b11
iff
b2,b3,b4,b5 are_COrtm_wrt b6,b7;