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

theorem
  for A, B being Matrix of n,REAL st n>0 holds for i,j st [i,j] in
  Indices (A*B) holds (A*B)*(i,j)=(A*(Col(B,j))).i
proof
  let A, B be Matrix of n,REAL;
  assume
A1: n>0;
  let i,j;
A2: width A=n by MATRIX_0:24;
  assume
A3: [i,j] in Indices (A*B);
  then i in dom (A*B) by ZFMISC_1:87;
  then
A4: i in Seg len (A*B) by FINSEQ_1:def 3;
A5: len B=n by MATRIX_0:def 2;
  then
A6: width A=len (Col(B,j)) by A2,MATRIX_0:def 8;
  width A=len B by A5,MATRIX_0:24;
  then
A7: (A*B)*(i,j) = Line(A,i) "*" Col(B,j) by A3,MATRPROB:39;
  len (A*B)=n & len A=n by MATRIX_0:24;
  then i in Seg len (A*(Col(B,j))) by A1,A4,A2,A6,MATRIXR1:61;
  hence thesis by A1,A2,A7,A6,MATRPROB:41;
end;
