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

theorem
  for a being Real st n>0 holds a*(0_Rmatrix(n,m)) = 0_Rmatrix(n,m)
proof
  let a be Real;
A1: len (a*(0_Rmatrix(n,m)))=len (0_Rmatrix(n,m)) & width (a*(0_Rmatrix(n,m)
  ))= width (0_Rmatrix(n,m)) by MATRIXR1:27;
  assume
A2: n>0;
  then
A3: width (0_Rmatrix(n,m))=m & len (0_Rmatrix(n,m))=n by MATRIX_0:23;
  a*(0_Rmatrix(n,m))=a*(0_Rmatrix(n,m)+0_Rmatrix(n,m)) by A2,Th65
    .=a*(0_Rmatrix(n,m)) +a*(0_Rmatrix(n,m)) by A3,MATRIXR1:43;
  hence thesis by A3,A1,MATRIXR1:37;
end;
