let n be Element of NAT ; :: thesis: for A, B being Matrix of n, REAL
for i, j being Element of 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 Element of NAT st [i,j] in Indices (B * A) holds
(B * A) * i,j = ((Line B,i) * A) . j
let i, j be Element of 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: ( i in dom (B * A) & j in Seg (width (B * A)) ) by ZFMISC_1:106;
A3: len A = n by MATRIX_1:def 3;
A4: width B = n by MATRIX_1:25;
A5: width B = len A by A3, MATRIX_1:25;
A6: (B * A) * i,j = (Line B,i) "*" (Col A,j) by A1, A5, MATRPROB:39;
A7: len A = len (Line B,i) by A3, A4, MATRIX_1:def 8;
width A = width (B * A) by A3, A4, MATRPROB:39;
then j in Seg (len ((Line B,i) * A)) by A2, A7, MATRIXR1:62;
hence (B * A) * i,j = ((Line B,i) * A) . j by A6, A7, MATRPROB:40; :: thesis: verum