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_1:def 9;
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 (Col B,i) = len B by MATRIX_1:def 9;
A7: dom A = dom B by A1, FINSEQ_3:31;
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_1:106;
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 A7, A9, MATRIX_1:def 9;
then ((Col A,i) . j) - ((Col B,i) . j) = (A - B) * j,i by A1, A2, A5, 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_1:146;
hence Col (A - B),i = (Col A,i) - (Col B,i) by A4, A3, A8, MATRIX_1:def 9; :: thesis: verum