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

theorem Th59:
  for x being FinSequence of REAL, A,B being Matrix of REAL st len
  x=width B & width A=len B & len x >0 & len B>0 holds (A*B)*x=A*(B*x)
proof
  let x be FinSequence of REAL,A,B be Matrix of REAL;
  assume that
A1: len x=width B and
A2: width A=len B and
A3: len x >0 and
A4: len B>0;
  len ColVec2Mx x=len x by A3,MATRIXR1:def 9;
  hence (A*B)*x = Col(A*(B*(ColVec2Mx x)),1) by A1,A2,MATRIX_3:33
    .=A*(B*x) by A1,A3,A4,Th58;
end;
