let D be non empty set ; :: thesis: for G being Matrix of D
for k, m being Nat st width G = m + 1 & m > 0 & k in Seg m holds
( Col ((DelCol (G,(width G))),k) = Col (G,k) & k in Seg (width (DelCol (G,(width G)))) )
let G be Matrix of D; :: thesis: for k, m being Nat st width G = m + 1 & m > 0 & k in Seg m holds
( Col ((DelCol (G,(width G))),k) = Col (G,k) & k in Seg (width (DelCol (G,(width G)))) )
let k, m be Nat; :: thesis: ( width G = m + 1 & m > 0 & k in Seg m implies ( Col ((DelCol (G,(width G))),k) = Col (G,k) & k in Seg (width (DelCol (G,(width G)))) ) )
assume that
A1: width G = m + 1 and
A2: m > 0 and
A3: k in Seg m ; :: thesis: ( Col ((DelCol (G,(width G))),k) = Col (G,k) & k in Seg (width (DelCol (G,(width G)))) )
k <= m by A3, FINSEQ_1:1;
then A4: k < width G by A1, NAT_1:13;
1 <= width G by A1, A2, SEQM_3:43;
then A5: width G in Seg (width G) by FINSEQ_1:1;
1 <= k by A3, FINSEQ_1:1;
hence ( Col ((DelCol (G,(width G))),k) = Col (G,k) & k in Seg (width (DelCol (G,(width G)))) ) by A1, A2, A5, A4, Th67; :: thesis: verum