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 - M2) by Def1;
A4: width ((M1 - M2)*') = width (M1 - M2) by Def1;
A5: width (M1 - M2) = width M1 by Th12;
A6: width (M1*') = width M1 by Def1;
A7: len (M1*') = len M1 by Def1;
A8: width (M2*') = width M2 by Def1;
A9: len (M1 - M2) = len M1 by Th12;
A10: len (M2*') = len M2 by Def1;
A11: now
    let i,j;
    assume
A12: [i,j] in Indices (M1-M2)*';
    then
A13: 1<=j by Th1;
A14: 1<= i by A12,Th1;
A15: j<=width (M1-M2) by A4,A12,Th1;
    then
A16: j<=width (M1*') by A5,Def1;
A17: i<=len (M1-M2) by A3,A12,Th1;
    then i<=len (M1*') by A9,Def1;
    then
A18: [i,j] in Indices (M1*') by A13,A14,A16,Th1;
A19: 1<= i by A12,Th1;
    then
A20: [i,j] in Indices M1 by A9,A5,A17,A13,A15,Th1;
A21: [i,j] in Indices M2 by A1,A2,A9,A5,A19,A17,A13,A15,Th1;
    [i,j] in Indices (M1-M2) by A19,A17,A13,A15,Th1;
    then ((M1-M2)*')*(i,j) = ((M1-M2)*(i,j))*' by Def1;
    hence (M1-M2)*'*(i,j) = (M1*(i,j)-M2*(i,j))*' by A1,A2,A20,Th13
      .= (M1*(i,j))*'-(M2*(i,j))*' by COMPLEX1:34
      .= M1*'*(i,j)-(M2*(i,j))*' by A20,Def1
      .= (M1*')*(i,j)-(M2*')*(i,j) by A21,Def1
      .= (M1*'-M2*')*(i,j) by A1,A2,A7,A10,A6,A8,A18,Th13;
  end;
A22: width (M1*') = width M1 by Def1;
A23: width (M1*' - M2*') = width (M1*') by Th12;
  len (M1*' - M2*') = len (M1*') by Th12;
  hence thesis by A3,A7,A9,A4,A5,A23,A22,A11,MATRIX_0:21;
end;
