let K be Field; :: thesis: for j being Nat
for A, B being Matrix of K st len A = len B & ex i being Nat st [i,j] in Indices A holds
Col ((A + B),j) = (Col (A,j)) + (Col (B,j))

let j be Nat; :: thesis: for A, B being Matrix of K st len A = len B & ex i being Nat st [i,j] in Indices A holds
Col ((A + B),j) = (Col (A,j)) + (Col (B,j))

let A, B be Matrix of K; :: thesis: ( len A = len B & ex i being Nat st [i,j] in Indices A implies Col ((A + B),j) = (Col (A,j)) + (Col (B,j)) )
A1: len (A + B) = len A by MATRIX_3:def 3;
assume A2: len A = len B ; :: thesis: ( for i being Nat holds not [i,j] in Indices A or Col ((A + B),j) = (Col (A,j)) + (Col (B,j)) )
then reconsider a = Col (A,j), b = Col (B,j) as Element of (len A) -tuples_on the carrier of K ;
given i being Nat such that A3: [i,j] in Indices A ; :: thesis: Col ((A + B),j) = (Col (A,j)) + (Col (B,j))
A4: width (A + B) = width A by MATRIX_3:def 3;
then A5: Indices (A + B) = Indices A by ;
A6: for k being Nat st 1 <= k & k <= len (Col ((A + B),j)) holds
(Col ((A + B),j)) . k = ((Col (A,j)) + (Col (B,j))) . k
proof
let k be Nat; :: thesis: ( 1 <= k & k <= len (Col ((A + B),j)) implies (Col ((A + B),j)) . k = ((Col (A,j)) + (Col (B,j))) . k )
assume A7: ( 1 <= k & k <= len (Col ((A + B),j)) ) ; :: thesis: (Col ((A + B),j)) . k = ((Col (A,j)) + (Col (B,j))) . k
A8: len (Col ((A + B),j)) = len A by ;
then k in Seg (len A) by ;
then A9: k in dom (A + B) by ;
len (Col (B,j)) = len B by MATRIX_0:def 8;
then k in Seg (len (Col (B,j))) by ;
then k in dom (Col (B,j)) by FINSEQ_1:def 3;
then reconsider e = (Col (B,j)) . k as Element of K by FINSEQ_2:11;
A10: dom A = Seg (len A) by FINSEQ_1:def 3
.= dom B by ;
A11: len (Col (A,j)) = len A by MATRIX_0:def 8;
then A12: k in Seg (len (Col (A,j))) by ;
then k in dom (Col (A,j)) by FINSEQ_1:def 3;
then reconsider d = (Col (A,j)) . k as Element of K by FINSEQ_2:11;
len ((Col (A,j)) + (Col (B,j))) = len (a + b)
.= len A by CARD_1:def 7
.= len (Col (A,j)) by CARD_1:def 7 ;
then k in dom ((Col (A,j)) + (Col (B,j))) by ;
then A13: ((Col (A,j)) + (Col (B,j))) . k = d + e by FVSUM_1:17;
j in Seg (width (A + B)) by ;
then A14: [k,j] in Indices (A + B) by ;
A15: (Col ((A + B),j)) . k = (A + B) * (k,j) by
.= (A * (k,j)) + (B * (k,j)) by ;
A16: k in dom A by ;
then (Col (A,j)) . k = A * (k,j) by MATRIX_0:def 8;
hence (Col ((A + B),j)) . k = ((Col (A,j)) + (Col (B,j))) . k by ; :: thesis: verum
end;
A17: len ((Col (A,j)) + (Col (B,j))) = len (a + b)
.= len A by CARD_1:def 7 ;
len (Col ((A + B),j)) = len A by ;
hence Col ((A + B),j) = (Col (A,j)) + (Col (B,j)) by ; :: thesis: verum