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

theorem Th31:
  for A being Matrix of REAL st len A=n & width A=m holds -A+A= 0_Rmatrix(n,m)
proof
  let A be Matrix of REAL;
  assume
A1: len A=n & width A=m;
  len (-(MXR2MXF A)) =len A & width (-(MXR2MXF A)) = width A by MATRIX_3:def 2;
  hence -A+A = MXF2MXR ((MXR2MXF A)+(-(MXR2MXF A))) by MATRIX_3:2
    .=0_Rmatrix(n,m) by A1,MATRIX_4:2;
end;
