reserve i,j for Nat;

theorem
  for A,B being Matrix of REAL st len A=len B & width A=width B & A = A
  + B holds B = 0_Rmatrix(len A,width A)
proof
  let A,B be Matrix of REAL;
  assume that
A1: len A=len B & width A=width B and
A2: A = A + B;
  0_Rmatrix(len A,width A)=A+B+(-A) by A2,MATRIX_4:2
    .=MXF2MXR ((MXR2MXF B)+(MXR2MXF A)+ -(MXR2MXF A)) by A1,MATRIX_3:2
    .=MXF2MXR ((MXR2MXF B)+(MXR2MXF A) -(MXR2MXF A)) by MATRIX_4:def 1
    .=B by A1,MATRIX_4:21;
  hence thesis;
end;
