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

theorem Th9:
  for i,j being Nat,M being Matrix of COMPLEX st [i,j] in Indices
  M holds (-M)*(i,j)= -(M*(i,j))
proof
  let i,j be Nat,M be Matrix of COMPLEX;
A1: COMPLEX2Field(-M) = COMPLEX2Field Field2COMPLEX (-COMPLEX2Field M) by
MATRIX_5:def 4
    .= -(COMPLEX2Field M) by MATRIX_5:6;
  reconsider m=COMPLEX2Field(-M) as Matrix of COMPLEX by COMPLFLD:def 1;
  reconsider m1=COMPLEX2Field M as Matrix of COMPLEX by COMPLFLD:def 1;
A2: 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
A3: [i,j] in Indices COMPLEX2Field M by MATRIX_5:def 1;
  (-M)*(i,j) = m*(i,j) by MATRIX_5:def 1
    .= (-(COMPLEX2Field M))*(i,j) by A1,COMPLFLD:def 1
    .= -((COMPLEX2Field M)*(i,j)) by A3,MATRIX_3:def 2;
  hence thesis by A2,COMPLFLD:2;
end;
