Article ANALMETR, MML version 4.99.1005
:: ANALMETR:prednot 1 => ANALMETR:pred 1
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;
pred Gen A2,A3 means
(for b1 being Element of the carrier of a1 holds
ex b2, b3 being Element of REAL st
b1 = (b2 * a2) + (b3 * a3)) &
(for b1, b2 being Element of REAL
st (b1 * a2) + (b2 * a3) = 0. a1
holds b1 = 0 & b2 = 0);
end;
:: ANALMETR:dfs 1
definiens
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;
To prove
Gen a2,a3
it is sufficient to prove
thus (for b1 being Element of the carrier of a1 holds
ex b2, b3 being Element of REAL st
b1 = (b2 * a2) + (b3 * a3)) &
(for b1, b2 being Element of REAL
st (b1 * a2) + (b2 * a3) = 0. a1
holds b1 = 0 & b2 = 0);
:: ANALMETR:def 1
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
Gen b2,b3
iff
(for b4 being Element of the carrier of b1 holds
ex b5, b6 being Element of REAL st
b4 = (b5 * b2) + (b6 * b3)) &
(for b4, b5 being Element of REAL
st (b4 * b2) + (b5 * b3) = 0. b1
holds b4 = 0 & b5 = 0);
:: ANALMETR:prednot 2 => ANALMETR:pred 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5 be Element of the carrier of a1;
pred A2,A3 are_Ort_wrt A4,A5 means
ex b1, b2, b3, b4 being Element of REAL st
a2 = (b1 * a4) + (b2 * a5) & a3 = (b3 * a4) + (b4 * a5) & (b1 * b3) + (b2 * b4) = 0;
end;
:: ANALMETR:dfs 2
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
a2,a3 are_Ort_wrt a4,a5
it is sufficient to prove
thus ex b1, b2, b3, b4 being Element of REAL st
a2 = (b1 * a4) + (b2 * a5) & a3 = (b3 * a4) + (b4 * a5) & (b1 * b3) + (b2 * b4) = 0;
:: ANALMETR:def 2
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 are_Ort_wrt b4,b5
iff
ex b6, b7, b8, b9 being Element of REAL st
b2 = (b6 * b4) + (b7 * b5) & b3 = (b8 * b4) + (b9 * b5) & (b6 * b8) + (b7 * b9) = 0;
:: ANALMETR: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 b4,b5
holds b2,b3 are_Ort_wrt b4,b5
iff
for b6, b7, b8, b9 being Element of REAL
st b2 = (b6 * b4) + (b7 * b5) & b3 = (b8 * b4) + (b9 * b5)
holds (b6 * b8) + (b7 * b9) = 0;
:: ANALMETR:th 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 holds
b2,b3 are_Ort_wrt b2,b3;
:: ANALMETR:th 7
theorem
ex b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct st
ex b2, b3 being Element of the carrier of b1 st
Gen b2,b3;
:: ANALMETR:th 8
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 b2,b3 are_Ort_wrt b4,b5
holds b3,b2 are_Ort_wrt b4,b5;
:: ANALMETR:th 9
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 being Element of the carrier of b1 holds
b4,0. b1 are_Ort_wrt b2,b3 & 0. b1,b5 are_Ort_wrt b2,b3;
:: ANALMETR: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
for b6, b7 being Element of REAL
st b2,b3 are_Ort_wrt b4,b5
holds b6 * b2,b7 * b3 are_Ort_wrt b4,b5;
:: ANALMETR: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
for b6, b7 being Element of REAL
st b2,b3 are_Ort_wrt b4,b5
holds b6 * b2,b3 are_Ort_wrt b4,b5 & b2,b7 * b3 are_Ort_wrt b4,b5;
:: ANALMETR:th 12
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 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
b4,b5 are_Ort_wrt b2,b3 & b5 <> 0. b1;
:: ANALMETR:th 13
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
st Gen b2,b3 & b4,b5 are_Ort_wrt b2,b3 & b4,b6 are_Ort_wrt b2,b3 & b4 <> 0. b1
holds ex b7, b8 being Element of REAL st
b7 * b5 = b8 * b6 & (b7 = 0 implies b8 <> 0);
:: ANALMETR:th 14
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
st Gen b2,b3 & b4,b5 are_Ort_wrt b2,b3 & b4,b6 are_Ort_wrt b2,b3
holds b4,b5 + b6 are_Ort_wrt b2,b3 & b4,b5 - b6 are_Ort_wrt b2,b3;
:: ANALMETR: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 & b4,b4 are_Ort_wrt b2,b3
holds b4 = 0. b1;
:: ANALMETR:th 16
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
st Gen b2,b3 & b4,b5 - b6 are_Ort_wrt b2,b3 & b5,b6 - b4 are_Ort_wrt b2,b3
holds b6,b4 - b5 are_Ort_wrt b2,b3;
:: ANALMETR:th 17
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 & b4 <> 0. b1
holds ex b6 being Element of REAL st
b5 - (b6 * b4),b4 are_Ort_wrt b2,b3;
:: ANALMETR:th 18
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 // b4,b5 or b2,b3 // b5,b4)
iff
ex b6, b7 being Element of REAL st
b6 * (b3 - b2) = b7 * (b5 - b4) &
(b6 = 0 implies b7 <> 0);
:: ANALMETR:th 19
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],[b4,b5]] in lambda DirPar b1
iff
ex b6, b7 being Element of REAL st
b6 * (b3 - b2) = b7 * (b5 - b4) &
(b6 = 0 implies b7 <> 0);
:: ANALMETR:prednot 3 => ANALMETR:pred 3
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 A2,A3,A4,A5 are_Ort_wrt A6,A7 means
a3 - a2,a5 - a4 are_Ort_wrt a6,a7;
end;
:: ANALMETR: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
a2,a3,a4,a5 are_Ort_wrt a6,a7
it is sufficient to prove
thus a3 - a2,a5 - a4 are_Ort_wrt a6,a7;
:: ANALMETR: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
b2,b3,b4,b5 are_Ort_wrt b6,b7
iff
b3 - b2,b5 - b4 are_Ort_wrt b6,b7;
:: ANALMETR:funcnot 1 => ANALMETR:func 1
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 Orthogonality(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_Ort_wrt a2,a3;
end;
:: ANALMETR:def 4
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 = Orthogonality(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_Ort_wrt b2,b3;
:: ANALMETR:th 22
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
the carrier of Lambda OASpace b1 = the carrier of b1;
:: ANALMETR:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
the CONGR of Lambda OASpace b1 = lambda DirPar b1;
:: ANALMETR: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
for b6, b7, b8, b9 being Element of the carrier of Lambda OASpace b1
st b6 = b2 & b7 = b3 & b8 = b4 & b9 = b5
holds b6,b7 // b8,b9
iff
ex b10, b11 being Element of REAL st
b10 * (b3 - b2) = b11 * (b5 - b4) &
(b10 = 0 implies b11 <> 0);
:: ANALMETR:structnot 1 => ANALMETR:struct 1
definition
struct(AffinStruct) ParOrtStr(#
carrier -> set,
CONGR -> Relation of [:the carrier of it,the carrier of it:],[:the carrier of it,the carrier of it:],
orthogonality -> Relation of [:the carrier of it,the carrier of it:],[:the carrier of it,the carrier of it:]
#);
end;
:: ANALMETR:attrnot 1 => ANALMETR:attr 1
definition
let a1 be ParOrtStr;
attr a1 is strict;
end;
:: ANALMETR:exreg 1
registration
cluster strict ParOrtStr;
end;
:: ANALMETR:aggrnot 1 => ANALMETR:aggr 1
definition
let a1 be set;
let a2, a3 be Relation of [:a1,a1:],[:a1,a1:];
aggr ParOrtStr(#a1,a2,a3#) -> strict ParOrtStr;
end;
:: ANALMETR:selnot 1 => ANALMETR:sel 1
definition
let a1 be ParOrtStr;
sel the orthogonality of a1 -> Relation of [:the carrier of a1,the carrier of a1:],[:the carrier of a1,the carrier of a1:];
end;
:: ANALMETR:exreg 2
registration
cluster non empty ParOrtStr;
end;
:: ANALMETR:prednot 4 => ANALMETR:pred 4
definition
let a1 be non empty ParOrtStr;
let a2, a3, a4, a5 be Element of the carrier of a1;
pred A2,A3 _|_ A4,A5 means
[[a2,a3],[a4,a5]] in the orthogonality of a1;
end;
:: ANALMETR:dfs 5
definiens
let a1 be non empty ParOrtStr;
let a2, a3, a4, a5 be Element of the carrier of a1;
To prove
a2,a3 _|_ a4,a5
it is sufficient to prove
thus [[a2,a3],[a4,a5]] in the orthogonality of a1;
:: ANALMETR:def 6
theorem
for b1 being non empty ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2,b3 _|_ b4,b5
iff
[[b2,b3],[b4,b5]] in the orthogonality of b1;
:: ANALMETR:funcnot 2 => ANALMETR:func 2
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 AMSpace(A1,A2,A3) -> strict ParOrtStr equals
ParOrtStr(#the carrier of a1,lambda DirPar a1,Orthogonality(a1,a2,a3)#);
end;
:: ANALMETR: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
AMSpace(b1,b2,b3) = ParOrtStr(#the carrier of b1,lambda DirPar b1,Orthogonality(b1,b2,b3)#);
:: ANALMETR: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 AMSpace(a1,a2,a3) -> non empty strict;
end;
:: ANALMETR:th 28
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
the carrier of AMSpace(b1,b2,b3) = the carrier of b1 &
the CONGR of AMSpace(b1,b2,b3) = lambda DirPar b1 &
the orthogonality of AMSpace(b1,b2,b3) = Orthogonality(b1,b2,b3);
:: ANALMETR:funcnot 3 => ANALMETR:func 3
definition
let a1 be non empty ParOrtStr;
func Af A1 -> strict AffinStruct equals
AffinStruct(#the carrier of a1,the CONGR of a1#);
end;
:: ANALMETR:def 8
theorem
for b1 being non empty ParOrtStr holds
Af b1 = AffinStruct(#the carrier of b1,the CONGR of b1#);
:: ANALMETR:funcreg 2
registration
let a1 be non empty ParOrtStr;
cluster Af a1 -> non empty strict;
end;
:: ANALMETR:th 30
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
Af AMSpace(b1,b2,b3) = Lambda OASpace b1;
:: ANALMETR: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
for b8, b9, b10, b11 being Element of the carrier of AMSpace(b1,b6,b7)
st b8 = b2 & b9 = b3 & b10 = b4 & b11 = b5
holds b8,b10 _|_ b9,b11
iff
b2,b4,b3,b5 are_Ort_wrt b6,b7;
:: ANALMETR: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 being Element of the carrier of b1
for b8, b9, b10, b11 being Element of the carrier of AMSpace(b1,b2,b3)
st b8 = b4 & b9 = b5 & b10 = b6 & b11 = b7
holds b8,b9 // b10,b11
iff
ex b12, b13 being Element of REAL st
b12 * (b5 - b4) = b13 * (b7 - b6) &
(b12 = 0 implies b13 <> 0);
:: ANALMETR:th 33
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, b5, b6, b7 being Element of the carrier of AMSpace(b1,b2,b3)
st b4,b5 _|_ b6,b7
holds b6,b7 _|_ b4,b5;
:: ANALMETR:th 34
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, b5, b6, b7 being Element of the carrier of AMSpace(b1,b2,b3)
st b4,b5 _|_ b6,b7
holds b4,b5 _|_ b7,b6;
:: ANALMETR:th 35
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 AMSpace(b1,b2,b3) holds
b4,b5 _|_ b6,b6;
:: ANALMETR:th 36
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, b5, b6, b7, b8, b9 being Element of the carrier of AMSpace(b1,b2,b3)
st b4,b5 _|_ b6,b7 & b4,b5 // b8,b9 & b4 <> b5
holds b6,b7 _|_ b8,b9;
:: ANALMETR:th 37
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 AMSpace(b1,b2,b3) holds
ex b7 being Element of the carrier of AMSpace(b1,b2,b3) st
b4,b5 _|_ b6,b7 & b6 <> b7;
:: ANALMETR: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
for b4, b5, b6, b7, b8, b9 being Element of the carrier of AMSpace(b1,b2,b3)
st Gen b2,b3 & b4,b5 _|_ b6,b7 & b4,b5 _|_ b8,b9 & b4 <> b5
holds b6,b7 // b8,b9;
:: ANALMETR: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
for b4, b5, b6, b7, b8 being Element of the carrier of AMSpace(b1,b2,b3)
st Gen b2,b3 & b4,b5 _|_ b6,b7 & b4,b5 _|_ b6,b8
holds b4,b5 _|_ b7,b8;
:: ANALMETR: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
for b4, b5 being Element of the carrier of AMSpace(b1,b2,b3)
st Gen b2,b3 & b4,b5 _|_ b4,b5
holds b4 = b5;
:: ANALMETR: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
for b4, b5, b6, b7 being Element of the carrier of AMSpace(b1,b2,b3)
st Gen b2,b3 & b4,b5 _|_ b6,b7 & b6,b5 _|_ b7,b4
holds b7,b5 _|_ b4,b6;
:: ANALMETR:th 42
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, b5 being Element of the carrier of AMSpace(b1,b2,b3)
st Gen b2,b3 & b4 <> b5
for b6 being Element of the carrier of AMSpace(b1,b2,b3) holds
ex b7 being Element of the carrier of AMSpace(b1,b2,b3) st
b4,b5 // b4,b7 & b4,b5 _|_ b7,b6;
:: ANALMETR:attrnot 2 => ANALMETR:attr 2
definition
let a1 be non empty ParOrtStr;
attr a1 is OrtAfSp-like means
AffinStruct(#the carrier of a1,the CONGR of a1#) is non empty non trivial AffinSpace-like AffinStruct &
(for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
(b1,b2 _|_ b1,b2 implies b1 = b2) &
b1,b2 _|_ b3,b3 &
(b1,b2 _|_ b3,b4 implies b1,b2 _|_ b4,b3 & b3,b4 _|_ b1,b2) &
(b1,b2 _|_ b5,b6 & b1,b2 // b7,b8 & not b5,b6 _|_ b7,b8 implies b1 = b2) &
(b1,b2 _|_ b5,b6 & b1,b2 _|_ b5,b8 implies b1,b2 _|_ b6,b8)) &
(for b1, b2, b3 being Element of the carrier of a1
st b1 <> b2
holds ex b4 being Element of the carrier of a1 st
b1,b2 // b1,b4 & b1,b2 _|_ b4,b3) &
(for b1, b2, b3 being Element of the carrier of a1 holds
ex b4 being Element of the carrier of a1 st
b1,b2 _|_ b3,b4 & b3 <> b4);
end;
:: ANALMETR:dfs 8
definiens
let a1 be non empty ParOrtStr;
To prove
a1 is OrtAfSp-like
it is sufficient to prove
thus AffinStruct(#the carrier of a1,the CONGR of a1#) is non empty non trivial AffinSpace-like AffinStruct &
(for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
(b1,b2 _|_ b1,b2 implies b1 = b2) &
b1,b2 _|_ b3,b3 &
(b1,b2 _|_ b3,b4 implies b1,b2 _|_ b4,b3 & b3,b4 _|_ b1,b2) &
(b1,b2 _|_ b5,b6 & b1,b2 // b7,b8 & not b5,b6 _|_ b7,b8 implies b1 = b2) &
(b1,b2 _|_ b5,b6 & b1,b2 _|_ b5,b8 implies b1,b2 _|_ b6,b8)) &
(for b1, b2, b3 being Element of the carrier of a1
st b1 <> b2
holds ex b4 being Element of the carrier of a1 st
b1,b2 // b1,b4 & b1,b2 _|_ b4,b3) &
(for b1, b2, b3 being Element of the carrier of a1 holds
ex b4 being Element of the carrier of a1 st
b1,b2 _|_ b3,b4 & b3 <> b4);
:: ANALMETR:def 9
theorem
for b1 being non empty ParOrtStr holds
b1 is OrtAfSp-like
iff
AffinStruct(#the carrier of b1,the CONGR of b1#) is non empty non trivial AffinSpace-like AffinStruct &
(for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1 holds
(b2,b3 _|_ b2,b3 implies b2 = b3) &
b2,b3 _|_ b4,b4 &
(b2,b3 _|_ b4,b5 implies b2,b3 _|_ b5,b4 & b4,b5 _|_ b2,b3) &
(b2,b3 _|_ b6,b7 & b2,b3 // b8,b9 & not b6,b7 _|_ b8,b9 implies b2 = b3) &
(b2,b3 _|_ b6,b7 & b2,b3 _|_ b6,b9 implies b2,b3 _|_ b7,b9)) &
(for b2, b3, b4 being Element of the carrier of b1
st b2 <> b3
holds ex b5 being Element of the carrier of b1 st
b2,b3 // b2,b5 & b2,b3 _|_ b5,b4) &
(for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
b2,b3 _|_ b4,b5 & b4 <> b5);
:: ANALMETR:exreg 3
registration
cluster non empty strict OrtAfSp-like ParOrtStr;
end;
:: ANALMETR:modenot 1
definition
mode OrtAfSp is non empty OrtAfSp-like ParOrtStr;
end;
:: ANALMETR:th 44
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 AMSpace(b1,b2,b3) is non empty OrtAfSp-like ParOrtStr;
:: ANALMETR:attrnot 3 => ANALMETR:attr 3
definition
let a1 be non empty ParOrtStr;
attr a1 is OrtAfPl-like means
AffinStruct(#the carrier of a1,the CONGR of a1#) is non empty non trivial AffinSpace-like 2-dimensional AffinStruct &
(for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
(b1,b2 _|_ b1,b2 implies b1 = b2) &
b1,b2 _|_ b3,b3 &
(b1,b2 _|_ b3,b4 implies b1,b2 _|_ b4,b3 & b3,b4 _|_ b1,b2) &
(b1,b2 _|_ b5,b6 & b1,b2 // b7,b8 & not b5,b6 _|_ b7,b8 implies b1 = b2) &
(b1,b2 _|_ b5,b6 & b1,b2 _|_ b7,b8 & not b5,b6 // b7,b8 implies b1 = b2)) &
(for b1, b2, b3 being Element of the carrier of a1 holds
ex b4 being Element of the carrier of a1 st
b1,b2 _|_ b3,b4 & b3 <> b4);
end;
:: ANALMETR:dfs 9
definiens
let a1 be non empty ParOrtStr;
To prove
a1 is OrtAfPl-like
it is sufficient to prove
thus AffinStruct(#the carrier of a1,the CONGR of a1#) is non empty non trivial AffinSpace-like 2-dimensional AffinStruct &
(for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
(b1,b2 _|_ b1,b2 implies b1 = b2) &
b1,b2 _|_ b3,b3 &
(b1,b2 _|_ b3,b4 implies b1,b2 _|_ b4,b3 & b3,b4 _|_ b1,b2) &
(b1,b2 _|_ b5,b6 & b1,b2 // b7,b8 & not b5,b6 _|_ b7,b8 implies b1 = b2) &
(b1,b2 _|_ b5,b6 & b1,b2 _|_ b7,b8 & not b5,b6 // b7,b8 implies b1 = b2)) &
(for b1, b2, b3 being Element of the carrier of a1 holds
ex b4 being Element of the carrier of a1 st
b1,b2 _|_ b3,b4 & b3 <> b4);
:: ANALMETR:def 10
theorem
for b1 being non empty ParOrtStr holds
b1 is OrtAfPl-like
iff
AffinStruct(#the carrier of b1,the CONGR of b1#) is non empty non trivial AffinSpace-like 2-dimensional AffinStruct &
(for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1 holds
(b2,b3 _|_ b2,b3 implies b2 = b3) &
b2,b3 _|_ b4,b4 &
(b2,b3 _|_ b4,b5 implies b2,b3 _|_ b5,b4 & b4,b5 _|_ b2,b3) &
(b2,b3 _|_ b6,b7 & b2,b3 // b8,b9 & not b6,b7 _|_ b8,b9 implies b2 = b3) &
(b2,b3 _|_ b6,b7 & b2,b3 _|_ b8,b9 & not b6,b7 // b8,b9 implies b2 = b3)) &
(for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
b2,b3 _|_ b4,b5 & b4 <> b5);
:: ANALMETR:exreg 4
registration
cluster non empty strict OrtAfPl-like ParOrtStr;
end;
:: ANALMETR:modenot 2
definition
mode OrtAfPl is non empty OrtAfPl-like ParOrtStr;
end;
:: ANALMETR:th 46
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 AMSpace(b1,b2,b3) is non empty OrtAfPl-like ParOrtStr;
:: ANALMETR:th 47
theorem
for b1 being non empty ParOrtStr
for b2 being set holds
b2 is Element of the carrier of b1
iff
b2 is Element of the carrier of Af b1;
:: ANALMETR:th 48
theorem
for b1 being non empty ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
for b6, b7, b8, b9 being Element of the carrier of Af b1
st b2 = b6 & b3 = b7 & b4 = b8 & b5 = b9
holds b2,b3 // b4,b5
iff
b6,b7 // b8,b9;
:: ANALMETR:funcreg 3
registration
let a1 be non empty OrtAfSp-like ParOrtStr;
cluster Af a1 -> non trivial strict AffinSpace-like;
end;
:: ANALMETR:funcreg 4
registration
let a1 be non empty OrtAfPl-like ParOrtStr;
cluster Af a1 -> non trivial strict AffinSpace-like 2-dimensional;
end;
:: ANALMETR:th 49
theorem
for b1 being non empty OrtAfPl-like ParOrtStr holds
b1 is non empty OrtAfSp-like ParOrtStr;
:: ANALMETR:condreg 1
registration
cluster non empty OrtAfPl-like -> OrtAfSp-like (ParOrtStr);
end;
:: ANALMETR:th 50
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
st Af b1 is non empty non trivial AffinSpace-like 2-dimensional AffinStruct
holds b1 is non empty OrtAfPl-like ParOrtStr;
:: ANALMETR:th 51
theorem
for b1 being non empty ParOrtStr holds
b1 is OrtAfPl-like
iff
(ex b2, b3 being Element of the carrier of b1 st
b2 <> b3) &
(for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1 holds
b2,b3 // b3,b2 &
b2,b3 // b4,b4 &
(b2,b3 // b6,b7 & b2,b3 // b8,b9 & not b6,b7 // b8,b9 implies b2 = b3) &
(b2,b3 // b2,b4 implies b3,b2 // b3,b4) &
(ex b10 being Element of the carrier of b1 st
b2,b3 // b4,b10 & b2,b4 // b3,b10) &
(ex b10, b11, b12 being Element of the carrier of b1 st
not b10,b11 // b10,b12) &
(ex b10 being Element of the carrier of b1 st
b2,b3 // b4,b10 & b4 <> b10) &
(b2,b3 // b3,b5 & b3 <> b2 implies ex b10 being Element of the carrier of b1 st
b4,b3 // b3,b10 & b4,b2 // b5,b10) &
(b2,b3 _|_ b2,b3 implies b2 = b3) &
b2,b3 _|_ b4,b4 &
(b2,b3 _|_ b4,b5 implies b2,b3 _|_ b5,b4 & b4,b5 _|_ b2,b3) &
(b2,b3 _|_ b6,b7 & b2,b3 // b8,b9 & not b6,b7 _|_ b8,b9 implies b2 = b3) &
(b2,b3 _|_ b6,b7 & b2,b3 _|_ b8,b9 & not b6,b7 // b8,b9 implies b2 = b3) &
(ex b10 being Element of the carrier of b1 st
b2,b3 _|_ b4,b10 & b4 <> b10) &
(not b2,b3 // b4,b5 implies ex b10 being Element of the carrier of b1 st
b2,b3 // b2,b10 & b4,b5 // b4,b10));
:: ANALMETR:prednot 5 => ANALMETR:pred 5
definition
let a1 be non empty ParOrtStr;
let a2, a3, a4 be Element of the carrier of a1;
pred LIN A2,A3,A4 means
a2,a3 // a2,a4;
end;
:: ANALMETR:dfs 10
definiens
let a1 be non empty ParOrtStr;
let a2, a3, a4 be Element of the carrier of a1;
To prove
LIN a2,a3,a4
it is sufficient to prove
thus a2,a3 // a2,a4;
:: ANALMETR:def 11
theorem
for b1 being non empty ParOrtStr
for b2, b3, b4 being Element of the carrier of b1 holds
LIN b2,b3,b4
iff
b2,b3 // b2,b4;
:: ANALMETR:funcnot 4 => ANALMETR:func 4
definition
let a1 be non empty ParOrtStr;
let a2, a3 be Element of the carrier of a1;
func Line(A2,A3) -> Element of bool the carrier of a1 means
for b1 being Element of the carrier of a1 holds
b1 in it
iff
LIN a2,a3,b1;
end;
:: ANALMETR:def 12
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
b4 = Line(b2,b3)
iff
for b5 being Element of the carrier of b1 holds
b5 in b4
iff
LIN b2,b3,b5;
:: ANALMETR:attrnot 4 => ANALMETR:attr 4
definition
let a1 be non empty ParOrtStr;
let a2 be Element of bool the carrier of a1;
attr a2 is being_line means
ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a2 = Line(b1,b2);
end;
:: ANALMETR:dfs 12
definiens
let a1 be non empty ParOrtStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is being_line
it is sufficient to prove
thus ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a2 = Line(b1,b2);
:: ANALMETR:def 13
theorem
for b1 being non empty ParOrtStr
for b2 being Element of bool the carrier of b1 holds
b2 is being_line(b1)
iff
ex b3, b4 being Element of the carrier of b1 st
b3 <> b4 & b2 = Line(b3,b4);
:: ANALMETR:prednot 6 => ANALMETR:attr 4
notation
let a1 be non empty ParOrtStr;
let a2 be Element of bool the carrier of a1;
synonym a2 is_line for being_line;
end;
:: ANALMETR:th 55
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4 being Element of the carrier of b1
for b5, b6, b7 being Element of the carrier of Af b1
st b2 = b5 & b3 = b6 & b4 = b7
holds LIN b2,b3,b4
iff
LIN b5,b6,b7;
:: ANALMETR:th 56
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of the carrier of Af b1
st b2 = b4 & b3 = b5
holds Line(b2,b3) = Line(b4,b5);
:: ANALMETR:th 57
theorem
for b1 being non empty ParOrtStr
for b2 being set holds
b2 is Element of bool the carrier of b1
iff
b2 is Element of bool the carrier of Af b1;
:: ANALMETR:th 58
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3 being Element of bool the carrier of Af b1
st b2 = b3
holds b2 is being_line(b1)
iff
b3 is being_line(Af b1);
:: ANALMETR:prednot 7 => ANALMETR:pred 6
definition
let a1 be non empty ParOrtStr;
let a2, a3 be Element of the carrier of a1;
let a4 be Element of bool the carrier of a1;
pred A2,A3 _|_ A4 means
ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a4 = Line(b1,b2) & a2,a3 _|_ b1,b2;
end;
:: ANALMETR:dfs 13
definiens
let a1 be non empty ParOrtStr;
let a2, a3 be Element of the carrier of a1;
let a4 be Element of bool the carrier of a1;
To prove
a2,a3 _|_ a4
it is sufficient to prove
thus ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a4 = Line(b1,b2) & a2,a3 _|_ b1,b2;
:: ANALMETR:def 14
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
b2,b3 _|_ b4
iff
ex b5, b6 being Element of the carrier of b1 st
b5 <> b6 & b4 = Line(b5,b6) & b2,b3 _|_ b5,b6;
:: ANALMETR:prednot 8 => ANALMETR:pred 7
definition
let a1 be non empty ParOrtStr;
let a2, a3 be Element of bool the carrier of a1;
pred A2 _|_ A3 means
ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a2 = Line(b1,b2) & b1,b2 _|_ a3;
end;
:: ANALMETR:dfs 14
definiens
let a1 be non empty ParOrtStr;
let a2, a3 be Element of bool the carrier of a1;
To prove
a2 _|_ a3
it is sufficient to prove
thus ex b1, b2 being Element of the carrier of a1 st
b1 <> b2 & a2 = Line(b1,b2) & b1,b2 _|_ a3;
:: ANALMETR:def 15
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of bool the carrier of b1 holds
b2 _|_ b3
iff
ex b4, b5 being Element of the carrier of b1 st
b4 <> b5 & b2 = Line(b4,b5) & b4,b5 _|_ b3;
:: ANALMETR:prednot 9 => ANALMETR:pred 8
definition
let a1 be non empty ParOrtStr;
let a2, a3 be Element of bool the carrier of a1;
pred A2 // A3 means
ex b1, b2, b3, b4 being Element of the carrier of a1 st
b1 <> b2 & b3 <> b4 & a2 = Line(b1,b2) & a3 = Line(b3,b4) & b1,b2 // b3,b4;
end;
:: ANALMETR:dfs 15
definiens
let a1 be non empty ParOrtStr;
let a2, a3 be Element of bool the carrier of a1;
To prove
a2 // a3
it is sufficient to prove
thus ex b1, b2, b3, b4 being Element of the carrier of a1 st
b1 <> b2 & b3 <> b4 & a2 = Line(b1,b2) & a3 = Line(b3,b4) & b1,b2 // b3,b4;
:: ANALMETR:def 16
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of bool the carrier of b1 holds
b2 // b3
iff
ex b4, b5, b6, b7 being Element of the carrier of b1 st
b4 <> b5 & b6 <> b7 & b2 = Line(b4,b5) & b3 = Line(b6,b7) & b4,b5 // b6,b7;
:: ANALMETR:th 62
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of the carrier of b1
for b4, b5 being Element of bool the carrier of b1 holds
(b2,b3 _|_ b4 implies b4 is being_line(b1)) & (b4 _|_ b5 implies b4 is being_line(b1) & b5 is being_line(b1));
:: ANALMETR:th 63
theorem
for b1 being non empty ParOrtStr
for b2, b3 being Element of bool the carrier of b1 holds
b2 _|_ b3
iff
ex b4, b5, b6, b7 being Element of the carrier of b1 st
b4 <> b5 & b6 <> b7 & b2 = Line(b4,b5) & b3 = Line(b6,b7) & b4,b5 _|_ b6,b7;
:: ANALMETR:th 64
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of bool the carrier of Af b1
st b2 = b4 & b3 = b5
holds b2 // b3
iff
b4 // b5;
:: ANALMETR:th 65
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b2 is being_line(b1)
holds b3,b3 _|_ b2;
:: ANALMETR:th 66
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
st b3,b4 _|_ b2 & (b3,b4 // b5,b6 or b5,b6 // b3,b4) & b3 <> b4
holds b5,b6 _|_ b2;
:: ANALMETR:th 67
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3,b4 _|_ b2
holds b4,b3 _|_ b2;
:: ANALMETR:prednot 10 => ANALMETR:pred 9
definition
let a1 be non empty OrtAfSp-like ParOrtStr;
let a2, a3 be Element of bool the carrier of a1;
redefine pred a2 // a3;
symmetry;
:: for a1 being non empty OrtAfSp-like ParOrtStr
:: for a2, a3 being Element of bool the carrier of a1
:: st a2 // a3
:: holds a3 // a2;
end;
:: ANALMETR:th 69
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5 being Element of the carrier of b1
st b4,b5 _|_ b2 & b2 // b3
holds b4,b5 _|_ b3;
:: ANALMETR:th 71
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2 & b3,b4 _|_ b2
holds b3 = b4;
:: ANALMETR:prednot 11 => ANALMETR:pred 10
definition
let a1 be non empty OrtAfSp-like ParOrtStr;
let a2, a3 be Element of bool the carrier of a1;
redefine pred a2 _|_ a3;
symmetry;
:: for a1 being non empty OrtAfSp-like ParOrtStr
:: for a2, a3 being Element of bool the carrier of a1
:: st a2 _|_ a3
:: holds a3 _|_ a2;
irreflexivity;
:: for a1 being non empty OrtAfSp-like ParOrtStr
:: for a2 being Element of bool the carrier of a1 holds
:: not a2 _|_ a2;
end;
:: ANALMETR:th 73
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4 being Element of bool the carrier of b1
st b2 _|_ b3 & b2 // b4
holds b4 _|_ b3;
:: ANALMETR:th 75
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
st b3 in b2 & b4 in b2 & b5,b6 _|_ b2
holds b5,b6 _|_ b3,b4 & b3,b4 _|_ b5,b6;
:: ANALMETR:th 76
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2 & b3 <> b4 & b2 is being_line(b1)
holds b2 = Line(b3,b4);
:: ANALMETR:th 77
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
st b3 in b2 & b4 in b2 & b3 <> b4 & b2 is being_line(b1) & (b3,b4 _|_ b5,b6 or b5,b6 _|_ b3,b4)
holds b5,b6 _|_ b2;
:: ANALMETR:th 78
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5, b6, b7 being Element of the carrier of b1
st b4 in b2 & b5 in b2 & b6 in b3 & b7 in b3 & b2 _|_ b3
holds b4,b5 _|_ b6,b7;
:: ANALMETR:th 79
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4, b5 being Element of bool the carrier of b1
for b6, b7, b8 being Element of the carrier of b1
st b6 in b2 & b6 in b3 & b7 in b2 & b8 in b3 & b7 <> b8 & b7 in b4 & b8 in b4 & b5 _|_ b2 & b5 _|_ b3 & b4 is being_line(b1)
holds b5 _|_ b4;
:: ANALMETR:th 80
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4 being Element of the carrier of b1 holds
b2,b3 _|_ b4,b4 & b4,b4 _|_ b2,b3 & b2,b3 // b4,b4 & b4,b4 // b2,b3;
:: ANALMETR:th 81
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 // b4,b5
holds b2,b3 // b5,b4 & b3,b2 // b4,b5 & b3,b2 // b5,b4 & b4,b5 // b2,b3 & b4,b5 // b3,b2 & b5,b4 // b2,b3 & b5,b4 // b3,b2;
:: ANALMETR:th 82
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st b2 <> b3 &
((b2,b3 // b4,b5 implies not b2,b3 // b6,b7) & (b2,b3 // b4,b5 implies not b6,b7 // b2,b3) & (b4,b5 // b2,b3 implies not b6,b7 // b2,b3) implies b4,b5 // b2,b3 & b2,b3 // b6,b7)
holds b4,b5 // b6,b7;
:: ANALMETR:th 83
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 _|_ b4,b5
holds b2,b3 _|_ b5,b4 & b3,b2 _|_ b4,b5 & b3,b2 _|_ b5,b4 & b4,b5 _|_ b2,b3 & b4,b5 _|_ b3,b2 & b5,b4 _|_ b2,b3 & b5,b4 _|_ b3,b2;
:: ANALMETR:th 84
theorem
for b1 being non empty OrtAfSp-like ParOrtStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st b2 <> b3 &
((b2,b3 // b4,b5 implies not b2,b3 _|_ b6,b7) & (b2,b3 // b6,b7 implies not b2,b3 _|_ b4,b5) & (b2,b3 // b4,b5 implies not b6,b7 _|_ b2,b3) & (b2,b3 // b6,b7 implies not b4,b5 _|_ b2,b3) & (b4,b5 // b2,b3 implies not b6,b7 _|_ b2,b3) & (b6,b7 // b2,b3 implies not b4,b5 _|_ b2,b3) & (b4,b5 // b2,b3 implies not b2,b3 _|_ b6,b7) implies b6,b7 // b2,b3 & b2,b3 _|_ b4,b5)
holds b4,b5 _|_ b6,b7;
:: ANALMETR:th 85
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5, b6, b7 being Element of the carrier of b1
st b2 <> b3 &
((b2,b3 _|_ b4,b5 implies not b2,b3 _|_ b6,b7) & (b2,b3 _|_ b4,b5 implies not b6,b7 _|_ b2,b3) & (b4,b5 _|_ b2,b3 implies not b6,b7 _|_ b2,b3) implies b4,b5 _|_ b2,b3 & b2,b3 _|_ b6,b7)
holds b4,b5 // b6,b7;
:: ANALMETR:th 86
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3 being Element of bool the carrier of b1
for b4, b5, b6, b7 being Element of the carrier of b1
st b4 in b2 & b5 in b2 & b4 <> b5 & b2 is being_line(b1) & b6 in b3 & b7 in b3 & b6 <> b7 & b3 is being_line(b1) & b4,b5 // b6,b7
holds b2 // b3;
:: ANALMETR:th 87
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4 being Element of bool the carrier of b1
st b2 _|_ b3 & b4 _|_ b3
holds b2 // b4;
:: ANALMETR:th 88
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3 being Element of bool the carrier of b1
st b2 _|_ b3
holds ex b4 being Element of the carrier of b1 st
b4 in b2 & b4 in b3;
:: ANALMETR:th 89
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4, b5 being Element of the carrier of b1
st b2,b3 _|_ b4,b5
holds ex b6 being Element of the carrier of b1 st
LIN b2,b3,b6 & LIN b4,b5,b6;
:: ANALMETR:th 90
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
st b3,b4 _|_ b2
holds ex b5 being Element of the carrier of b1 st
LIN b3,b4,b5 & b5 in b2;
:: ANALMETR:th 91
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2, b3, b4 being Element of the carrier of b1 holds
ex b5 being Element of the carrier of b1 st
b2,b5 _|_ b3,b4 & LIN b3,b4,b5;
:: ANALMETR:th 92
theorem
for b1 being non empty OrtAfPl-like ParOrtStr
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b2 is being_line(b1)
holds ex b4 being Element of the carrier of b1 st
b3,b4 _|_ b2 & b4 in b2;