reserve i,j for Nat;

theorem
  for a being Real, x being FinSequence of REAL,
      A being Matrix of REAL
  st len A=len x & len x>0 & width A>0 holds (a*x)*A=a*(x*A)
proof
  let a be Real,x be FinSequence of REAL,A be Matrix of REAL;
  assume that
A1: len A=len x and
A2: len x>0 and
A3: width A>0;
A4: (A@)*x=x*A by A1,A2,A3,Th52;
A5: width (A@)=len x by A1,A3,MATRIX_0:54;
  then
A6: (A@)*(a*x)=a*((A@)*x) by A2,Th59;
A7: len (a*x)=len x by RVSUM_1:117;
  len (A@)>0 by A3,MATRIX_0:54;
  then (a*x)*((A@)@)=(A@)*(a*x) by A2,A5,A7,Th53;
  hence thesis by A1,A2,A3,A6,A4,MATRIX_0:57;
end;
