let n, m be Nat; :: thesis: for M being Matrix of n,m,F_Real
for A being linearly-independent Subset of (REAL-NS n) st the_rank_of M = n holds
(Mx2Tran M) .: A is linearly-independent
let M be Matrix of n,m,F_Real; :: thesis: for A being linearly-independent Subset of (REAL-NS n) st the_rank_of M = n holds
(Mx2Tran M) .: A is linearly-independent
let A be linearly-independent Subset of (REAL-NS n); :: thesis: ( the_rank_of M = n implies (Mx2Tran M) .: A is linearly-independent )
assume A1: the_rank_of M = n ; :: thesis: (Mx2Tran M) .: A is linearly-independent
reconsider A0 = A as linearly-independent Subset of (TOP-REAL n) by Th4, Th28;
(Mx2Tran M) .: A0 is linearly-independent by A1, MATRTOP2:23;
hence (Mx2Tran M) .: A is linearly-independent ; :: thesis: verum