reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th3:
  for i,j being Nat,a being Complex,M being Matrix of
  COMPLEX st [i,j] in Indices M holds (a*M)*(i,j) = a*(M*(i,j))
proof
  let i,j be Nat,a be Complex,M be Matrix of COMPLEX;
  reconsider m1=COMPLEX2Field M as Matrix of COMPLEX by COMPLFLD:def 1;
A1: M*(i,j) = m1*(i,j) by MATRIX_5:def 1
    .= COMPLEX2Field(M)*(i,j) by COMPLFLD:def 1;
  assume [i,j] in Indices M;
  then
A2: [i,j] in Indices COMPLEX2Field M by MATRIX_5:def 1;
  a in COMPLEX by XCMPLX_0:def 2;
  then reconsider aa=a as Element of F_Complex by COMPLFLD:def 1;
  reconsider m=COMPLEX2Field(a*M) as Matrix of COMPLEX by COMPLFLD:def 1;
A3: COMPLEX2Field(a*M) = COMPLEX2Field Field2COMPLEX (aa*(COMPLEX2Field M))
  by MATRIX_5:def 7
    .= aa*(COMPLEX2Field M) by MATRIX_5:6;
  (a*M)*(i,j)= m*(i,j) by MATRIX_5:def 1
    .= (aa*(COMPLEX2Field M))*(i,j) by A3,COMPLFLD:def 1
    .= aa*((COMPLEX2Field M)*(i,j)) by A2,MATRIX_3:def 5
    .= a*(COMPLEX2Field (M)*(i,j));
  hence thesis by A1;
end;
