let A, B be Matrix of REAL; :: thesis: ( len A = len B & width A = width B implies for i being Nat st 1 <= i & i <= width A holds
Col ((A - B),i) = (Col (A,i)) - (Col (B,i)) )
assume that
A1: len A = len B and
A2: width A = width B ; :: thesis: for i being Nat st 1 <= i & i <= width A holds
Col ((A - B),i) = (Col (A,i)) - (Col (B,i))
A3: len (A - B) = len A by A1, A2, Th6;
let i be Nat; :: thesis: ( 1 <= i & i <= width A implies Col ((A - B),i) = (Col (A,i)) - (Col (B,i)) )
A4: len (Col (A,i)) = len A by MATRIX_0:def 8;
assume ( 1 <= i & i <= width A ) ; :: thesis: Col ((A - B),i) = (Col (A,i)) - (Col (B,i))
then A5: i in Seg (width A) ;
A6: len (Col (B,i)) = len B by MATRIX_0:def 8;
A7: dom A = dom B by A1, FINSEQ_3:29;
A8: 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 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 A3, FINSEQ_1:def 3;
then A10: [j,i] in Indices A by A5, ZFMISC_1:87;
reconsider j = j as Nat ;
( (Col (A,i)) . j = A * (j,i) & (Col (B,i)) . j = B * (j,i) ) by A7, A9, MATRIX_0:def 8;
then ((Col (A,i)) . j) - ((Col (B,i)) . j) = (A - B) * (j,i) by A1, A2, A10, Th6;
hence ((Col (A,i)) - (Col (B,i))) . j = (A - B) * (j,i) by A1, A4, A6, Lm1; :: thesis: verum

end;

len ((Col (A,i)) - (Col (B,i))) = len (Col (A,i)) by A1, A4, A6, RVSUM_1:116;
hence Col ((A - B),i) = (Col (A,i)) - (Col (B,i)) by A4, A3, A8, MATRIX_0:def 8; :: thesis: verum