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

theorem Th4:
  for a being Complex,M being Matrix of COMPLEX holds (a*M)
  *' = a*'*(M*')
proof
  let a be Complex,M be Matrix of COMPLEX;
  reconsider aa = a as Element of COMPLEX by XCMPLX_0:def 2;
A1: len (a*M) = len M by Th2;
A2: width (a*M) = width M by Th2;
A3: width M = width (M*') by Def1;
A4: len ((a*M)*') = len (a*M) by Def1;
A5: width ((a*M)*') = width (a*M) by Def1;
A6: len M = len (M*') by Def1;
A7: now
    let i,j;
    assume
A8: [i,j] in Indices (a*M)*';
    then
A9: 1<= i by Th1;
A10: 1<=j by A8,Th1;
A11: j<=width (a*M) by A5,A8,Th1;
A12: i<=len (a*M) by A4,A8,Th1;
    then
A13: [i,j] in Indices M by A1,A2,A9,A10,A11,Th1;
A14: [i,j] in Indices (M*') by A1,A6,A2,A3,A9,A12,A10,A11,Th1;
    [i,j] in Indices (a*M) by A9,A12,A10,A11,Th1;
    then ((a*M)*')*(i,j) = ((a*M)*(i,j))*' by Def1;
    hence (a*M)*'*(i,j) = (aa*(M*(i,j)))*' by A13,Th3
      .= aa*'*(M*(i,j))*' by COMPLEX1:35
      .= a*'*((M*')*(i,j)) by A13,Def1
      .= (a*'*(M*'))*(i,j) by A14,Th3;
  end;
  len (a*'*(M*')) = len (M*') by Th2;
  then len (a*'*(M*')) = len M by Def1;
  then
A15: len ((a*M)*')=len (a*'*(M*')) by A4,Th2;
  width (a*'*(M*')) = width (M*') by Th2;
  then width (a*'*(M*')) = width M by Def1;
  hence thesis by A15,A5,A7,Th2,MATRIX_0:21;
end;
