let n be non zero Element of NAT ; :: thesis: ex B being finite Subset of () st
( B is Basis of () & card B = n & ex A being FinSequence of n -tuples_on BOOLEAN st
( len A = n & A is one-to-one & card (rng A) = n & rng A = B & ( 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 ) ) ) ) )

set V = n -BinaryVectSp ;
consider A being FinSequence of n -tuples_on BOOLEAN such that
A1: ( len A = n & A is one-to-one & card (rng A) = n & ( 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 ) ) ) ) by Th8;
reconsider B = rng A as finite Subset of () ;
A2: B is linearly-independent by ;
Lin B = ModuleStr(# the carrier of (), the addF of (), the ZeroF of (), the lmult of () #) by ;
then B is Basis of () by ;
hence ex B being finite Subset of () st
( B is Basis of () & card B = n & ex A being FinSequence of n -tuples_on BOOLEAN st
( len A = n & A is one-to-one & card (rng A) = n & rng A = B & ( 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 ) ) ) ) ) by A1; :: thesis: verum