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

theorem
  for M1,M2 being Matrix of COMPLEX st len M1=len M2 & width M1=width M2
  holds (M1 + M2)*' = M1*' + M2*'
proof
  let M1,M2 be Matrix of COMPLEX;
  assume that
A1: len M1=len M2 and
A2: width M1=width M2;
A3: len (M1 + M2) = len M1 by Th5;
A4: width ((M1 + M2)*') = width (M1 + M2) by Def1;
A5: width (M1 + M2) = width M1 by Th5;
A6: len ((M1 + M2)*') = len (M1 + M2) by Def1;
A7: now
    let i,j;
    assume
A8: [i,j] in Indices (M1+M2)*';
    then
A9: 1<=j by Th1;
A10: 1<= i by A8,Th1;
A11: j<=width (M1+M2) by A4,A8,Th1;
    then
A12: j<=width (M1*') by A5,Def1;
A13: i<=len (M1+M2) by A6,A8,Th1;
    then i<=len (M1*') by A3,Def1;
    then
A14: [i,j] in Indices (M1*') by A9,A10,A12,Th1;
A15: 1<= i by A8,Th1;
    then
A16: [i,j] in Indices M1 by A3,A5,A13,A9,A11,Th1;
A17: [i,j] in Indices M2 by A1,A2,A3,A5,A15,A13,A9,A11,Th1;
    [i,j] in Indices (M1+M2) by A15,A13,A9,A11,Th1;
    then ((M1+M2)*')*(i,j) = ((M1+M2)*(i,j))*' by Def1;
    hence (M1+M2)*'*(i,j) = (M1*(i,j)+M2*(i,j))*' by A16,Th6
      .= (M1*(i,j))*'+(M2*(i,j))*' by COMPLEX1:32
      .= M1*'*(i,j)+(M2*(i,j))*' by A16,Def1
      .= (M1*')*(i,j)+(M2*')*(i,j) by A17,Def1
      .= (M1*'+M2*')*(i,j) by A14,Th6;
  end;
A18: width (M1*') = width M1 by Def1;
A19: width (M1*' + M2*') = width (M1*') by Th5;
A20: len (M1*' + M2*') = len (M1*') by Th5;
  len (M1*') = len M1 by Def1;
  hence thesis by A6,A3,A20,A4,A5,A19,A18,A7,MATRIX_0:21;
end;
