reserve i,j for Nat;

theorem
  for n,m being Nat, x being FinSequence of REAL st len x=n &
  n>0 holds x*(0_Rmatrix(n,m))=0*m
proof
  let n,m be Nat,x be FinSequence of REAL;
  assume that
A1: len x=n and
A2: n>0;
A3: len (0_Rmatrix(n,m))=n by A2,MATRIX_0:23;
  then
A4: len (x*(0_Rmatrix(n,m)))=width (0_Rmatrix(n,m)) by A1,Th62;
A5: width (0_Rmatrix(n,m))=m by A2,MATRIX_0:23;
  then x*(0_Rmatrix(n,m))=x*(0_Rmatrix(n,m)+0_Rmatrix(n,m)) by A2,A3,Th36
    .=x*(0_Rmatrix(n,m)) +x*(0_Rmatrix(n,m)) by A1,A5,A3,Th64;
  then
  0*m =x*(0_Rmatrix(n,m)) +x*(0_Rmatrix(n,m)) - x*(0_Rmatrix(n,m)) by A5,A4,Th3
;
  hence thesis by A4,Th14;
end;
