theorem :: MATRIX13:123

for m, n being Nat
for K being Field
for M being Matrix of m,n,K
for RANK being Element of NAT holds
( the_rank_of M = RANK iff ( ex U being finite Subset of (n -VectSp_over K) st
( U is linearly-independent & U c= lines M & card U = RANK ) & ( for W being finite Subset of (n -VectSp_over K) st W is linearly-independent & W c= lines M holds
card W <= RANK ) ) )