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

theorem Th6:
  for i,j being Nat,M1,M2 being Matrix of COMPLEX st [i,j] in
  Indices M1 holds (M1+M2)*(i,j) = M1*(i,j) + M2*(i,j)
proof
  let i,j be Nat,M1,M2 be Matrix of COMPLEX;
A1: COMPLEX2Field(M1+M2) = COMPLEX2Field Field2COMPLEX ((COMPLEX2Field M1)+(
  COMPLEX2Field M2)) by MATRIX_5:def 3
    .= (COMPLEX2Field M1)+(COMPLEX2Field M2) by MATRIX_5:6;
  reconsider m1=COMPLEX2Field M1, m2=COMPLEX2Field M2 as Matrix of COMPLEX by
COMPLFLD:def 1;
  set m=COMPLEX2Field(M1+M2);
  reconsider m9=m as Matrix of COMPLEX by COMPLFLD:def 1;
A2: M1*(i,j) = m1*(i,j) by MATRIX_5:def 1
    .=COMPLEX2Field(M1)*(i,j) by COMPLFLD:def 1;
  assume [i,j] in Indices M1;
  then
A3: [i,j] in Indices COMPLEX2Field M1 by MATRIX_5:def 1;
A4: M2*(i,j) = m2*(i,j) by MATRIX_5:def 1
    .=COMPLEX2Field(M2)*(i,j) by COMPLFLD:def 1;
  (M1+M2)*(i,j) = m9*(i,j) by MATRIX_5:def 1
    .= m*(i,j) by COMPLFLD:def 1
    .=(COMPLEX2Field M1)*(i,j)+(COMPLEX2Field M2)*(i,j) by A1,A3,MATRIX_3:def 3
;
  hence thesis by A2,A4;
end;
