reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th61:
  for A, B being Matrix of n,REAL holds for i,j st [i,j] in
  Indices (B*A) holds (B*A)*(i,j)=((Line(B,i))*A).j
proof
  let A,B be Matrix of n,REAL;
  let i,j;
  assume
A1: [i,j] in Indices (B*A);
  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 thesis by A4,A6,MATRPROB:40;
end;
