let A, B be Matrix of REAL ; :: thesis: ( len A = len B implies for i being Nat st 1 <= i & i <= width A holds
Col (A + B),i = (Col A,i) + (Col B,i) )
assume A1: len A = len B ; :: thesis: for i being Nat st 1 <= i & i <= width A holds
Col (A + B),i = (Col A,i) + (Col B,i)
then A2: dom A = dom B by FINSEQ_3:31;
let i be Nat; :: thesis: ( 1 <= i & i <= width A implies Col (A + B),i = (Col A,i) + (Col B,i) )
A3: len (Col A,i) = len A by MATRIX_1:def 9;
len (Col B,i) = len B by MATRIX_1:def 9;
then A4: len ((Col A,i) + (Col B,i)) = len (Col A,i) by A1, A3, RVSUM_1:145;
assume ( 1 <= i & i <= width A ) ; :: thesis: Col (A + B),i = (Col A,i) + (Col B,i)
then A5: i in Seg (width A) by FINSEQ_1:3;
A6: len (A + B) = len A by Th25;
Seg (len (A + B)) = dom (A + B) by FINSEQ_1:def 3;
then A7: dom ((Col A,i) + (Col B,i)) = dom (A + B) by A3, A6, A4, FINSEQ_1:def 3;
for j being Nat st j in dom (A + B) holds
((Col A,i) + (Col B,i)) . j = (A + B) * j,i

proof

let j be Nat; :: thesis: ( j in dom (A + B) implies ((Col A,i) + (Col B,i)) . j = (A + B) * j,i )
assume A8: j in dom (A + B) ; :: thesis: ((Col A,i) + (Col B,i)) . j = (A + B) * j,i
then j in Seg (len (A + B)) by FINSEQ_1:def 3;
then A9: j in dom A by A6, FINSEQ_1:def 3;
then A10: [j,i] in Indices A by A5, ZFMISC_1:106;
reconsider j = j as Element of NAT by ORDINAL1:def 13;
( (Col A,i) . j = A * j,i & (Col B,i) . j = B * j,i ) by A2, A9, MATRIX_1:def 9;
then ((Col A,i) . j) + ((Col B,i) . j) = (A + B) * j,i by A10, Th25;
hence ((Col A,i) + (Col B,i)) . j = (A + B) * j,i by A7, A8, VALUED_1:def 1; :: thesis: verum

end;

hence Col (A + B),i = (Col A,i) + (Col B,i) by A3, A6, A4, MATRIX_1:def 9; :: thesis: verum