let n be Nat; :: thesis: for A, B being Matrix of n,REAL
for i, j being Nat st [i,j] in Indices (B * A) holds
(B * A) * (i,j) = ((Line (B,i)) * A) . j
let A, B be Matrix of n,REAL; :: thesis: for i, j being Nat st [i,j] in Indices (B * A) holds
(B * A) * (i,j) = ((Line (B,i)) * A) . j
let i, j be Nat; :: thesis: ( [i,j] in Indices (B * A) implies (B * A) * (i,j) = ((Line (B,i)) * A) . j )
assume A1: [i,j] in Indices (B * A) ; :: thesis: (B * A) * (i,j) = ((Line (B,i)) * A) . j
then A2: j in Seg (width (B * A)) by ZFMISC_1:87;
A3: len A = n by MATRIX_0:def 2;
then width B = len A by MATRIX_0:24;
then A4: (B * A) * (i,j) = (Line (B,i)) "*" (Col (A,j)) by A1, MATRPROB:39;
A5: width B = n by MATRIX_0:24;
then A6: len A = len (Line (B,i)) by A3, MATRIX_0:def 7;
width A = width (B * A) by A3, A5, MATRPROB:39;
then j in Seg (len ((Line (B,i)) * A)) by A2, A6, MATRIXR1:62;
hence (B * A) * (i,j) = ((Line (B,i)) * A) . j by A4, A6, MATRPROB:40; :: thesis: verum