let n, m, k be Nat; :: thesis: for D being non empty set
for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg (width B) holds
Col (A ^^ B),((width A) + i) = Col B,i
let D be non empty set ; :: thesis: for A being Matrix of n,m,D
for B being Matrix of n,k,D
for i being Nat st i in Seg (width B) holds
Col (A ^^ B),((width A) + i) = Col B,i
let A be Matrix of n,m,D; :: thesis: for B being Matrix of n,k,D
for i being Nat st i in Seg (width B) holds
Col (A ^^ B),((width A) + i) = Col B,i
let B be Matrix of n,k,D; :: thesis: for i being Nat st i in Seg (width B) holds
Col (A ^^ B),((width A) + i) = Col B,i
let i be Nat; :: thesis: ( i in Seg (width B) implies Col (A ^^ B),((width A) + i) = Col B,i )
assume A1: i in Seg (width B) ; :: thesis: Col (A ^^ B),((width A) + i) = Col B,i
set AB = A ^^ B;
A2: len (A ^^ B) = n by MATRIX_1:def 3;
A3: len B = n by MATRIX_1:def 3;

now

A4: dom B = Seg n by A3, FINSEQ_1:def 3;
let j be Nat; :: thesis: ( j in Seg n implies (Col (A ^^ B),((width A) + i)) . j = (Col B,i) . j )
assume A5: j in Seg n ; :: thesis: (Col (A ^^ B),((width A) + i)) . j = (Col B,i) . j
n <> 0 by A5;
then width (A ^^ B) = (width A) + (width B) by MATRIX_1:24;
then A6: (width A) + i in Seg (width (A ^^ B)) by A1, FINSEQ_1:81;
A7: ( dom (Line B,j) = Seg (width B) & len (Line A,j) = width A ) by FINSEQ_1:def 18, FINSEQ_2:144;
dom (A ^^ B) = Seg n by A2, FINSEQ_1:def 3;
hence (Col (A ^^ B),((width A) + i)) . j = (A ^^ B) * j,((width A) + i) by A5, MATRIX_1:def 9
.= (Line (A ^^ B),j) . ((width A) + i) by A6, MATRIX_1:def 8
.= ((Line A,j) ^ (Line B,j)) . ((width A) + i) by A5, Th15
.= (Line B,j) . i by A1, A7, FINSEQ_1:def 7
.= B * j,i by A1, MATRIX_1:def 8
.= (Col B,i) . j by A5, A4, MATRIX_1:def 9 ;
:: thesis: verum

end;

hence Col (A ^^ B),((width A) + i) = Col B,i by A3, A2, FINSEQ_2:139; :: thesis: verum