theorem Th49: :: MATRIX11:49

for n being Nat
for K being commutative Ring
for i, j being Nat st i in Seg n & j in Seg n & i < j holds
for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds
(Path_product M) . q = - ((Path_product M) . p)