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

theorem Th13:
  for i,j being Nat,M1,M2 be Matrix of COMPLEX st len M1=len M2 &
width M1=width M2 & [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;
  assume that
A1: len M1 =len M2 and
A2: width M1=width M2 and
A3: [i,j] in Indices M1;
A4: j<=width M2 by A2,A3,Th1;
A5: 1<=j by A3,Th1;
A6: 1<= i by A3,Th1;
  i<=len M2 by A1,A3,Th1;
  then [i,j] in Indices M2 by A6,A5,A4,Th1;
  then
A7: [i,j] in Indices COMPLEX2Field M2 by MATRIX_5:def 1;
  reconsider m2=COMPLEX2Field M2 as Matrix of COMPLEX by COMPLFLD:def 1;
  reconsider m1=COMPLEX2Field M1 as Matrix of COMPLEX by COMPLFLD:def 1;
  set m=COMPLEX2Field(M1-M2);
A8: COMPLEX2Field(M1-M2) = COMPLEX2Field Field2COMPLEX ((COMPLEX2Field M1)-(
  COMPLEX2Field M2)) by MATRIX_5:def 5
    .= (COMPLEX2Field M1)-(COMPLEX2Field M2) by MATRIX_5:6;
  reconsider m9=m as Matrix of COMPLEX by COMPLFLD:def 1;
A9: M1*(i,j) = m1*(i,j) by MATRIX_5:def 1
    .= COMPLEX2Field(M1)*(i,j) by COMPLFLD:def 1;
A10: [i,j] in Indices COMPLEX2Field M1 by A3,MATRIX_5:def 1;
  M2*(i,j) = m2*(i,j) by MATRIX_5:def 1
    .= COMPLEX2Field(M2)*(i,j) by COMPLFLD:def 1;
  then
A11: -(M2*(i,j))=-(COMPLEX2Field(M2)*(i,j)) by COMPLFLD:2;
  (M1-M2)*(i,j) = m9*(i,j) by MATRIX_5:def 1
    .= m*(i,j) by COMPLFLD:def 1
    .= ((COMPLEX2Field M1)+(-(COMPLEX2Field M2)))*(i,j) by A8,MATRIX_4:def 1
    .= (COMPLEX2Field M1)*(i,j)+(-COMPLEX2Field M2)*(i,j) by A10,MATRIX_3:def 3
    .= (COMPLEX2Field M1)*(i,j)+-((COMPLEX2Field M2)*(i,j)) by A7,
MATRIX_3:def 2;
  hence thesis by A9,A11;
end;
