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

theorem Th4:
  for M being Matrix of REAL st [i,j] in Indices M holds (a*M)*(i,j
  )=a*(M*(i,j))
proof
  let M be Matrix of REAL;
  a in REAL by XREAL_0:def 1;
  then reconsider aa=a as Element of F_Real by VECTSP_1:def 5;
A1: MXR2MXF(a*M) = MXR2MXF MXF2MXR (aa*(MXR2MXF M)) by MATRIXR1:def 7
    .= aa*(MXR2MXF M);
  assume [i,j] in Indices M;
  then (a*M)*(i,j) = aa*((MXR2MXF M)*(i,j)) by A1,MATRIX_3:def 5,VECTSP_1:def 5
;
  hence thesis by VECTSP_1:def 5;
end;
