theorem Th68: :: MATRIX13:68

for m9, n9 being Nat
for K being Field
for M9 being Matrix of n9,m9,K
for P, Q being finite without_zero Subset of NAT
for i being Nat
for j0 being non zero Nat st j0 in Seg n9 & i in P & not j0 in P & card P = card Q & [:P,Q:] c= Indices M9 holds
( card P = card ((P \ {i}) \/ {j0}) & [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 & ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) )