reserve i,j for Nat;

theorem
  for n,m being Nat, x being FinSequence of REAL st len x=m &
  n>0 & m>0 holds (0_Rmatrix(n,m))*x=0*n
proof
  let n,m be Nat, x be FinSequence of REAL;
  assume that
A1: len x=m and
A2: n>0 and
A3: m>0;
A4: width 0_Rmatrix(n,m)=m by A2,MATRIX_0:23;
  then
A5: len ((0_Rmatrix(n,m))*x)=len (0_Rmatrix(n,m)) by A1,A3,Th61;
A6: len 0_Rmatrix(n,m) = n by A2,MATRIX_0:23;
  then (0_Rmatrix(n,m))*x=(0_Rmatrix(n,m)+0_Rmatrix(n,m))*x by A2,A4,Th36
    .=(0_Rmatrix(n,m))*x +(0_Rmatrix(n,m))*x by A1,A3,A6,A4,Th63;
  then
  0*n =(0_Rmatrix(n,m))*x +(0_Rmatrix(n,m))*x - (0_Rmatrix(n,m))*x by A6,A5,Th3
;
  hence thesis by A5,Th14;
end;
