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

theorem Th3:
  for a being Real, A being Matrix of REAL st [i,j] in
  Indices A holds (a*A)*(i,j) = a*(A*(i,j))
proof
  let a be Real,A be Matrix of REAL;
  assume
A1: [i,j] in Indices A;
  reconsider aa=a as Element of F_Real by XREAL_0:def 1;
  (a*A)*(i,j) = (MXF2MXR (aa*(MXR2MXF A)))*(i,j) by MATRIXR1:def 7
    .= aa*((MXR2MXF A)*(i,j)) by A1,MATRIX_3:def 5
    .= a*(A*(i,j));
  hence thesis;
end;
