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

theorem
  for M being Matrix of COMPLEX st width M > 0 holds (M@)*' =(M*')@
proof
  let M be Matrix of COMPLEX;
  assume
A1: width M > 0;
  width (M*') = width M by Def1;
  then
A2: width ((M*')@) = len (M*') by A1,MATRIX_0:54
    .= len M by Def1;
A3: width ((M@)*') = width (M@) by Def1;
A4: len ((M@)*') = len (M@) by Def1
    .= width M by MATRIX_0:def 6;
A5: width ((M@)*') = width (M@) by Def1
    .= len M by A1,MATRIX_0:54;
A6: len ((M@)*') = len (M@) by Def1;
A7: for i,j st [i,j] in Indices ((M@)*') holds ((M@)*')*(i,j) = ((M*')@)*(i, j)
  proof
    let i,j;
    assume
A8: [i,j] in Indices ((M@)*');
    then
A9: 1<= i by Th1;
A10: 1<=j by A8,Th1;
A11: j<=width (M@) by A3,A8,Th1;
    i<=len (M@) by A6,A8,Th1;
    then
A12: [i,j] in Indices (M@) by A9,A10,A11,Th1;
    then
A13: [j,i] in Indices (M) by MATRIX_0:def 6;
    j<=width ((M@)*') by A8,Th1;
    then
A14: j<=len (M*') by A5,Def1;
    i<=len ((M@)*') by A8,Th1;
    then
A15: i<=width (M*') by A4,Def1;
A16: 1<=i by A8,Th1;
    1<= j by A8,Th1;
    then
A17: [j,i] in Indices (M*') by A14,A16,A15,Th1;
    ((M@)*')*(i,j) = ((M@)*(i,j))*' by A12,Def1
      .= (M*(j,i))*' by A13,MATRIX_0:def 6
      .= ((M*'))*(j,i) by A13,Def1
      .= ((M*')@)*(i,j) by A17,MATRIX_0:def 6;
    hence thesis;
  end;
  len ((M*')@) = width (M*') by MATRIX_0:def 6
    .= width M by Def1;
  hence thesis by A4,A5,A2,A7,MATRIX_0:21;
end;
