Article ANALOAF, MML version 4.99.1005
:: ANALOAF:prednot 1 => ANALOAF:pred 1
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 // A4,A5 means
(a2 <> a3 & a4 <> a5) implies ex b1, b2 being Element of REAL st
0 < b1 &
0 < b2 &
b1 * (a3 - a2) = b2 * (a5 - a4);
end;
:: ANALOAF:dfs 1
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 // a4,a5
it is sufficient to prove
thus (a2 <> a3 & a4 <> a5) implies ex b1, b2 being Element of REAL st
0 < b1 &
0 < b2 &
b1 * (a3 - a2) = b2 * (a5 - a4);
:: ANALOAF:def 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 holds
b2,b3 // b4,b5
iff
(b2 <> b3 & b4 <> b5 implies ex b6, b7 being Element of REAL st
0 < b6 &
0 < b7 &
b6 * (b3 - b2) = b7 * (b5 - b4));
:: ANALOAF: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 holds
(b2 - b3) + (b3 - b4) = b2 - b4;
:: ANALOAF:th 9
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 = b4 + b5
holds b2 - b5 = b4 - b3;
:: ANALOAF:th 10
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 Element of REAL holds
b4 * (b2 - b3) = - (b4 * (b3 - b2));
:: ANALOAF:th 11
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 REAL holds
(b4 - b5) * (b2 - b3) = (b5 - b4) * (b3 - b2);
:: ANALOAF: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
for b4 being Element of REAL
st b4 <> 0 & b4 * b2 = b3
holds b2 = b4 " * b3;
:: ANALOAF:th 13
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 Element of REAL holds
(b4 <> 0 & b4 * b2 = b3 implies b2 = b4 " * b3) &
(b4 <> 0 & b2 = b4 " * b3 implies b4 * b2 = b3);
:: ANALOAF:th 16
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 // b4,b5 & b2 <> b3 & b4 <> b5
holds ex b6, b7 being Element of REAL st
b6 * (b3 - b2) = b7 * (b5 - b4) &
0 < b6 &
0 < b7;
:: ANALOAF:th 17
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 // b2,b3;
:: ANALOAF:th 18
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
b2,b3 // b4,b4 & b2,b2 // b3,b4;
:: ANALOAF:th 19
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 b2,b3 // b3,b2
holds b2 = b3;
:: ANALOAF:th 20
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 & b2,b3 // b4,b5 & b2,b3 // b6,b7
holds b4,b5 // b6,b7;
:: ANALOAF:th 21
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 // b4,b5
holds b3,b2 // b5,b4 & b4,b5 // b2,b3;
:: ANALOAF:th 22
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 b2,b3 // b3,b4
holds b2,b3 // b2,b4;
:: ANALOAF:th 23
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 b2,b3 // b2,b4 & not b2,b3 // b3,b4
holds b2,b4 // b4,b3;
:: ANALOAF: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 b2 - b3 = b4 - b5
holds b3,b2 // b5,b4;
:: ANALOAF: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
st b2 = (b3 + b4) - b5
holds b5,b3 // b4,b2 & b5,b4 // b3,b2;
:: ANALOAF:th 26
theorem
for 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
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 & b2,b4 // b3,b5 & b3 <> b5;
:: ANALOAF:th 27
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 & b3,b2 // b2,b4
holds ex b6 being Element of the carrier of b1 st
b5,b2 // b2,b6 & b5,b3 // b4,b6;
:: ANALOAF: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
st for b4, b5 being Element of REAL
st (b4 * b2) + (b5 * b3) = 0. b1
holds b4 = 0 & b5 = 0
holds b2 <> b3 & b2 <> 0. b1 & b3 <> 0. b1;
:: ANALOAF:th 29
theorem
for 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
for b4, b5 being Element of REAL
st (b4 * b2) + (b5 * b3) = 0. b1
holds b4 = 0 & b5 = 0
holds ex b2, b3, b4, b5 being Element of the carrier of b1 st
not b2,b3 // b4,b5 & not b2,b3 // b5,b4;
:: ANALOAF:th 31
theorem
for 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
for b4 being Element of the carrier of b1 holds
ex b5, b6 being Element of REAL st
(b5 * b2) + (b6 * b3) = b4
for b2, b3, b4, b5 being Element of the carrier of b1
st not b2,b3 // b4,b5 & not b2,b3 // b5,b4
holds ex b6 being Element of the carrier of b1 st
(b2,b3 // b2,b6 or b2,b3 // b6,b2) & (b4,b5 // b4,b6 or b4,b5 // b6,b4);
:: ANALOAF:structnot 1 => ANALOAF:struct 1
definition
struct(1-sorted) AffinStruct(#
carrier -> set,
CONGR -> Relation of [:the carrier of it,the carrier of it:],[:the carrier of it,the carrier of it:]
#);
end;
:: ANALOAF:attrnot 1 => ANALOAF:attr 1
definition
let a1 be AffinStruct;
attr a1 is strict;
end;
:: ANALOAF:exreg 1
registration
cluster strict AffinStruct;
end;
:: ANALOAF:aggrnot 1 => ANALOAF:aggr 1
definition
let a1 be set;
let a2 be Relation of [:a1,a1:],[:a1,a1:];
aggr AffinStruct(#a1,a2#) -> strict AffinStruct;
end;
:: ANALOAF:selnot 1 => ANALOAF:sel 1
definition
let a1 be AffinStruct;
sel the CONGR of a1 -> Relation of [:the carrier of a1,the carrier of a1:],[:the carrier of a1,the carrier of a1:];
end;
:: ANALOAF:exreg 2
registration
cluster non empty strict AffinStruct;
end;
:: ANALOAF:prednot 2 => ANALOAF:pred 2
definition
let a1 be non empty AffinStruct;
let a2, a3, a4, a5 be Element of the carrier of a1;
pred A2,A3 // A4,A5 means
[[a2,a3],[a4,a5]] in the CONGR of a1;
end;
:: ANALOAF:dfs 2
definiens
let a1 be non empty AffinStruct;
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 CONGR of a1;
:: ANALOAF:def 2
theorem
for b1 being non empty AffinStruct
for b2, b3, b4, b5 being Element of the carrier of b1 holds
b2,b3 // b4,b5
iff
[[b2,b3],[b4,b5]] in the CONGR of b1;
:: ANALOAF:funcnot 1 => ANALOAF:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func DirPar A1 -> 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;
end;
:: ANALOAF:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Relation of [:the carrier of b1,the carrier of b1:],[:the carrier of b1,the carrier of b1:] holds
b2 = DirPar b1
iff
for b3, b4 being set holds
[b3,b4] in b2
iff
ex b5, b6, b7, b8 being Element of the carrier of b1 st
b3 = [b5,b6] & b4 = [b7,b8] & b5,b6 // b7,b8;
:: ANALOAF:th 33
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 DirPar b1
iff
b2,b3 // b4,b5;
:: ANALOAF:funcnot 2 => ANALOAF:func 2
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
func OASpace A1 -> strict AffinStruct equals
AffinStruct(#the carrier of a1,DirPar a1#);
end;
:: ANALOAF:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
OASpace b1 = AffinStruct(#the carrier of b1,DirPar b1#);
:: ANALOAF:funcreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster OASpace a1 -> non empty strict;
end;
:: ANALOAF:th 35
theorem
for 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
for b4, b5 being Element of REAL
st (b4 * b2) + (b5 * b3) = 0. b1
holds b4 = 0 & b5 = 0
holds (ex b2, b3 being Element of the carrier of OASpace b1 st
b2 <> b3) &
(for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of OASpace b1 holds
b2,b3 // b4,b4 &
(b2,b3 // b3,b2 implies b2 = b3) &
(b2 <> b3 & b2,b3 // b6,b7 & b2,b3 // b8,b9 implies b6,b7 // b8,b9) &
(b2,b3 // b4,b5 implies b3,b2 // b5,b4) &
(b2,b3 // b3,b4 implies b2,b3 // b2,b4) &
(b2,b3 // b2,b4 & not b2,b3 // b3,b4 implies b2,b4 // b4,b3)) &
(ex b2, b3, b4, b5 being Element of the carrier of OASpace b1 st
not b2,b3 // b4,b5 & not b2,b3 // b5,b4) &
(for b2, b3, b4 being Element of the carrier of OASpace b1 holds
ex b5 being Element of the carrier of OASpace b1 st
b2,b3 // b4,b5 & b2,b4 // b3,b5 & b3 <> b5) &
(for b2, b3, b4, b5 being Element of the carrier of OASpace b1
st b2 <> b4 & b4,b2 // b2,b5
holds ex b6 being Element of the carrier of OASpace b1 st
b3,b2 // b2,b6 & b3,b4 // b5,b6);
:: ANALOAF:th 36
theorem
for 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
for b4 being Element of the carrier of b1 holds
ex b5, b6 being Element of REAL st
(b5 * b2) + (b6 * b3) = b4
for b2, b3, b4, b5 being Element of the carrier of OASpace b1
st not b2,b3 // b4,b5 & not b2,b3 // b5,b4
holds ex b6 being Element of the carrier of OASpace b1 st
(b2,b3 // b2,b6 or b2,b3 // b6,b2) & (b4,b5 // b4,b6 or b4,b5 // b6,b4);
:: ANALOAF:attrnot 2 => ANALOAF:attr 2
definition
let a1 be non empty AffinStruct;
attr a1 is OAffinSpace-like means
(for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
b1,b2 // b3,b3 &
(b1,b2 // b2,b1 implies b1 = b2) &
(b1 <> b2 & b1,b2 // b5,b6 & b1,b2 // b7,b8 implies b5,b6 // b7,b8) &
(b1,b2 // b3,b4 implies b2,b1 // b4,b3) &
(b1,b2 // b2,b3 implies b1,b2 // b1,b3) &
(b1,b2 // b1,b3 & not b1,b2 // b2,b3 implies b1,b3 // b3,b2)) &
(ex b1, b2, b3, b4 being Element of the carrier of a1 st
not b1,b2 // b3,b4 & not 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 & b1,b3 // b2,b4 & b2 <> b4) &
(for b1, b2, b3, b4 being Element of the carrier of a1
st b1 <> b3 & b3,b1 // b1,b4
holds ex b5 being Element of the carrier of a1 st
b2,b1 // b1,b5 & b2,b3 // b4,b5);
end;
:: ANALOAF:dfs 5
definiens
let a1 be non empty AffinStruct;
To prove
a1 is OAffinSpace-like
it is sufficient to prove
thus (for b1, b2, b3, b4, b5, b6, b7, b8 being Element of the carrier of a1 holds
b1,b2 // b3,b3 &
(b1,b2 // b2,b1 implies b1 = b2) &
(b1 <> b2 & b1,b2 // b5,b6 & b1,b2 // b7,b8 implies b5,b6 // b7,b8) &
(b1,b2 // b3,b4 implies b2,b1 // b4,b3) &
(b1,b2 // b2,b3 implies b1,b2 // b1,b3) &
(b1,b2 // b1,b3 & not b1,b2 // b2,b3 implies b1,b3 // b3,b2)) &
(ex b1, b2, b3, b4 being Element of the carrier of a1 st
not b1,b2 // b3,b4 & not 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 & b1,b3 // b2,b4 & b2 <> b4) &
(for b1, b2, b3, b4 being Element of the carrier of a1
st b1 <> b3 & b3,b1 // b1,b4
holds ex b5 being Element of the carrier of a1 st
b2,b1 // b1,b5 & b2,b3 // b4,b5);
:: ANALOAF:def 5
theorem
for b1 being non empty AffinStruct holds
b1 is OAffinSpace-like
iff
(for b2, b3, b4, b5, b6, b7, b8, b9 being Element of the carrier of b1 holds
b2,b3 // b4,b4 &
(b2,b3 // b3,b2 implies b2 = b3) &
(b2 <> b3 & b2,b3 // b6,b7 & b2,b3 // b8,b9 implies b6,b7 // b8,b9) &
(b2,b3 // b4,b5 implies b3,b2 // b5,b4) &
(b2,b3 // b3,b4 implies b2,b3 // b2,b4) &
(b2,b3 // b2,b4 & not b2,b3 // b3,b4 implies b2,b4 // b4,b3)) &
(ex b2, b3, b4, b5 being Element of the carrier of b1 st
not b2,b3 // b4,b5 & not 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 & b2,b4 // b3,b5 & b3 <> b5) &
(for b2, b3, b4, b5 being Element of the carrier of b1
st b2 <> b4 & b4,b2 // b2,b5
holds ex b6 being Element of the carrier of b1 st
b3,b2 // b2,b6 & b3,b4 // b5,b6);
:: ANALOAF:exreg 3
registration
cluster non empty non trivial strict OAffinSpace-like AffinStruct;
end;
:: ANALOAF:modenot 1
definition
mode OAffinSpace is non empty non trivial OAffinSpace-like AffinStruct;
end;
:: ANALOAF:th 37
theorem
for b1 being non empty AffinStruct holds
(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 // b4,b4 &
(b2,b3 // b3,b2 implies b2 = b3) &
(b2 <> b3 & b2,b3 // b6,b7 & b2,b3 // b8,b9 implies b6,b7 // b8,b9) &
(b2,b3 // b4,b5 implies b3,b2 // b5,b4) &
(b2,b3 // b3,b4 implies b2,b3 // b2,b4) &
(b2,b3 // b2,b4 & not b2,b3 // b3,b4 implies b2,b4 // b4,b3)) &
(ex b2, b3, b4, b5 being Element of the carrier of b1 st
not b2,b3 // b4,b5 & not 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 & b2,b4 // b3,b5 & b3 <> b5) &
(for b2, b3, b4, b5 being Element of the carrier of b1
st b2 <> b4 & b4,b2 // b2,b5
holds ex b6 being Element of the carrier of b1 st
b3,b2 // b2,b6 & b3,b4 // b5,b6)
iff
b1 is non empty non trivial OAffinSpace-like AffinStruct;
:: ANALOAF:th 38
theorem
for 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
for b4, b5 being Element of REAL
st (b4 * b2) + (b5 * b3) = 0. b1
holds b4 = 0 & b5 = 0
holds OASpace b1 is non empty non trivial OAffinSpace-like AffinStruct;
:: ANALOAF:attrnot 3 => ANALOAF:attr 3
definition
let a1 be non empty non trivial OAffinSpace-like AffinStruct;
attr a1 is 2-dimensional means
for b1, b2, b3, b4 being Element of the carrier of a1
st not b1,b2 // b3,b4 & not b1,b2 // b4,b3
holds ex b5 being Element of the carrier of a1 st
(b1,b2 // b1,b5 or b1,b2 // b5,b1) & (b3,b4 // b3,b5 or b3,b4 // b5,b3);
end;
:: ANALOAF:dfs 6
definiens
let a1 be non empty non trivial OAffinSpace-like AffinStruct;
To prove
a1 is 2-dimensional
it is sufficient to prove
thus for b1, b2, b3, b4 being Element of the carrier of a1
st not b1,b2 // b3,b4 & not b1,b2 // b4,b3
holds ex b5 being Element of the carrier of a1 st
(b1,b2 // b1,b5 or b1,b2 // b5,b1) & (b3,b4 // b3,b5 or b3,b4 // b5,b3);
:: ANALOAF:def 6
theorem
for b1 being non empty non trivial OAffinSpace-like AffinStruct holds
b1 is 2-dimensional
iff
for b2, b3, b4, b5 being Element of the carrier of b1
st not b2,b3 // b4,b5 & not b2,b3 // b5,b4
holds ex b6 being Element of the carrier of b1 st
(b2,b3 // b2,b6 or b2,b3 // b6,b2) & (b4,b5 // b4,b6 or b4,b5 // b6,b4);
:: ANALOAF:exreg 4
registration
cluster non empty non trivial strict OAffinSpace-like 2-dimensional AffinStruct;
end;
:: ANALOAF:modenot 2
definition
mode OAffinPlane is non empty non trivial OAffinSpace-like 2-dimensional AffinStruct;
end;
:: ANALOAF:th 50
theorem
for b1 being non empty AffinStruct holds
(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 // b4,b4 &
(b2,b3 // b3,b2 implies b2 = b3) &
(b2 <> b3 & b2,b3 // b6,b7 & b2,b3 // b8,b9 implies b6,b7 // b8,b9) &
(b2,b3 // b4,b5 implies b3,b2 // b5,b4) &
(b2,b3 // b3,b4 implies b2,b3 // b2,b4) &
(b2,b3 // b2,b4 & not b2,b3 // b3,b4 implies b2,b4 // b4,b3)) &
(ex b2, b3, b4, b5 being Element of the carrier of b1 st
not b2,b3 // b4,b5 & not 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 & b2,b4 // b3,b5 & b3 <> b5) &
(for b2, b3, b4, b5 being Element of the carrier of b1
st b2 <> b4 & b4,b2 // b2,b5
holds ex b6 being Element of the carrier of b1 st
b3,b2 // b2,b6 & b3,b4 // b5,b6) &
(for b2, b3, b4, b5 being Element of the carrier of b1
st not b2,b3 // b4,b5 & not b2,b3 // b5,b4
holds ex b6 being Element of the carrier of b1 st
(b2,b3 // b2,b6 or b2,b3 // b6,b2) & (b4,b5 // b4,b6 or b4,b5 // b6,b4))
iff
b1 is non empty non trivial OAffinSpace-like 2-dimensional AffinStruct;
:: ANALOAF:th 51
theorem
for 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
(for b4, b5 being Element of REAL
st (b4 * b2) + (b5 * b3) = 0. b1
holds b4 = 0 & b5 = 0) &
(for b4 being Element of the carrier of b1 holds
ex b5, b6 being Element of REAL st
b4 = (b5 * b2) + (b6 * b3))
holds OASpace b1 is non empty non trivial OAffinSpace-like 2-dimensional AffinStruct;