let n be non zero Element of NAT ; :: thesis: for A being FinSequence of n -tuples_on BOOLEAN
for B being finite Subset of (n -BinaryVectSp) st rng A = B & len A = n & A is one-to-one & ( for i, j being Nat st i in Seg n & j in Seg n holds
( ( i = j implies (A . i) . j = TRUE ) & ( i <> j implies (A . i) . j = FALSE ) ) ) holds
Lin B = ModuleStr(# the carrier of (n -BinaryVectSp), the addF of (n -BinaryVectSp), the ZeroF of (n -BinaryVectSp), the lmult of (n -BinaryVectSp) #)
let A be FinSequence of n -tuples_on BOOLEAN; :: thesis: for B being finite Subset of (n -BinaryVectSp) st rng A = B & len A = n & A is one-to-one & ( for i, j being Nat st i in Seg n & j in Seg n holds
( ( i = j implies (A . i) . j = TRUE ) & ( i <> j implies (A . i) . j = FALSE ) ) ) holds
Lin B = ModuleStr(# the carrier of (n -BinaryVectSp), the addF of (n -BinaryVectSp), the ZeroF of (n -BinaryVectSp), the lmult of (n -BinaryVectSp) #)
let B be finite Subset of (n -BinaryVectSp); :: thesis: ( rng A = B & len A = n & A is one-to-one & ( for i, j being Nat st i in Seg n & j in Seg n holds
( ( i = j implies (A . i) . j = TRUE ) & ( i <> j implies (A . i) . j = FALSE ) ) ) implies Lin B = ModuleStr(# the carrier of (n -BinaryVectSp), the addF of (n -BinaryVectSp), the ZeroF of (n -BinaryVectSp), the lmult of (n -BinaryVectSp) #) )
assume that
A1: rng A = B and
A2: len A = n and
A3: A is one-to-one and
A4: for i, j being Nat st i in Seg n & j in Seg n holds
( ( i = j implies (A . i) . j = TRUE ) & ( i <> j implies (A . i) . j = FALSE ) ) ; :: thesis: Lin B = ModuleStr(# the carrier of (n -BinaryVectSp), the addF of (n -BinaryVectSp), the ZeroF of (n -BinaryVectSp), the lmult of (n -BinaryVectSp) #)
set V = n -BinaryVectSp ;
for x being object holds
( x in the carrier of (Lin B) iff x in the carrier of (n -BinaryVectSp) )

proof

let x be object ; :: thesis: ( x in the carrier of (Lin B) iff x in the carrier of (n -BinaryVectSp) )

hereby :: thesis: ( x in the carrier of (n -BinaryVectSp) implies x in the carrier of (Lin B) )

assume A5: x in the carrier of (Lin B) ; :: thesis: x in the carrier of (n -BinaryVectSp)
the carrier of (Lin B) c= the carrier of (n -BinaryVectSp) by VECTSP_4:def 2;
hence x in the carrier of (n -BinaryVectSp) by A5; :: thesis: verum

end;

assume x in the carrier of (n -BinaryVectSp) ; :: thesis: x in the carrier of (Lin B)
then reconsider v = x as Element of n -tuples_on BOOLEAN ;
consider l being Linear_Combination of B such that
A6: for j being Nat st j in Seg n holds
v . j = l . (A . j) by Th11, A1, A2, A3;
reconsider Suml = Sum l as Element of n -tuples_on BOOLEAN ;
A7: len v = n by Lm1;
A8: len Suml = n by Lm1;
A9: dom v = Seg n by FINSEQ_1:def 3, A7
.= dom Suml by FINSEQ_1:def 3, A8 ;
for j being Nat st j in dom v holds
v . j = Suml . j

proof

let j be Nat; :: thesis: ( j in dom v implies v . j = Suml . j )
assume j in dom v ; :: thesis: v . j = Suml . j
then A10: j in Seg n by FINSEQ_1:def 3, A7;
thus v . j = l . (A . j) by A6, A10
.= Suml . j by Th9, A1, A2, A3, A4, A10 ; :: thesis: verum

end;

then x in Lin B by FINSEQ_1:13, A9, VECTSP_7:7;
hence x in the carrier of (Lin B) by STRUCT_0:def 5; :: thesis: verum

end;

hence Lin B = ModuleStr(# the carrier of (n -BinaryVectSp), the addF of (n -BinaryVectSp), the ZeroF of (n -BinaryVectSp), the lmult of (n -BinaryVectSp) #) by TARSKI:2, VECTSP_4:31; :: thesis: verum