reserve a,b for Real,
  i,j,n for Nat,
  M,M1,M2,M3,M4 for Matrix of n, REAL;

theorem Th3:
  for M1,M2 be Matrix of REAL st len M1=len M2 & width M1=width M2
  & [i,j] in Indices M1 holds (M1-M2)*(i,j)=(M1*(i,j)) - (M2*(i,j))
proof
  let M1,M2 be Matrix of REAL;
  assume that
A1: len M1=len M2 and
A2: width M1=width M2 and
A3: [i,j] in Indices M1;
A4: 1<=j & j<=width M2 by A2,A3,MATRIXC1:1;
  1<= i & i<=len M2 by A1,A3,MATRIXC1:1;
  then
A5: [i,j] in Indices MXR2MXF M2 by A4,MATRIXC1:1;
  (M1-M2)*(i,j) = ((MXR2MXF M1)+(-(MXR2MXF M2)))*(i,j) by MATRIX_4:def 1
,VECTSP_1:def 5
    .= (MXR2MXF M1)*(i,j)+(-MXR2MXF M2)*(i,j) by A3,MATRIX_3:def 3
    .= (MXR2MXF M1)*(i,j)+-((MXR2MXF M2)*(i,j)) by A5,MATRIX_3:def 2;
  hence thesis by VECTSP_1:def 5;
end;
