let D be non empty set ; :: thesis: for G being Matrix of D
for k, n, m being Nat st width G = m + 1 & m > 0 & k in Seg m & n in dom G holds
( (DelCol (G,1)) * (n,k) = G * (n,(k + 1)) & 1 in Seg (width G) )
let G be Matrix of D; :: thesis: for k, n, m being Nat st width G = m + 1 & m > 0 & k in Seg m & n in dom G holds
( (DelCol (G,1)) * (n,k) = G * (n,(k + 1)) & 1 in Seg (width G) )
let k, n, m be Nat; :: thesis: ( width G = m + 1 & m > 0 & k in Seg m & n in dom G implies ( (DelCol (G,1)) * (n,k) = G * (n,(k + 1)) & 1 in Seg (width G) ) )
assume that
A1: width G = m + 1 and
A2: m > 0 and
A3: k in Seg m and
A4: n in dom G ; :: thesis: ( (DelCol (G,1)) * (n,k) = G * (n,(k + 1)) & 1 in Seg (width G) )
1 <= width G by A1, A2, SEQM_3:43;
then A5: 1 in Seg (width G) by FINSEQ_1:1;
( 1 <= k & k <= m ) by A3, FINSEQ_1:1;
hence ( (DelCol (G,1)) * (n,k) = G * (n,(k + 1)) & 1 in Seg (width G) ) by A1, A4, A5, Th70; :: thesis: verum