reserve i,j for Element of NAT,
  x,y,z for FinSequence of COMPLEX,
  c for Element of COMPLEX,
  R,R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th13:
  for a being Complex holds (a*x)*' = a*'*(x*')
proof
  let a be Complex;
  reconsider aa = a as Element of COMPLEX by XCMPLX_0:def 2;
  len (a*'*(x*')) = len (x*') by Th3; then
A1: len (a*'*(x*')) = len x by Def1;
A2: len (a*x) = len x by Th3;
A3: now
    let i be Nat;
    assume that
A4: 1 <= i and
A5: i <= len ((a*x)*');
A6: i <= len (a*x) by A5,Def1;
    hence (a*x)*'.i = (((a*x).i)*') by A4,Def1
      .= (aa*(x.i))*' by Th12
      .= aa*'*(x.i)*' by COMPLEX1:35
      .= a*'*((x*').i) by A2,A4,A6,Def1
      .= (a*'*(x*')).i by Th12;
  end;
  len ((a*x)*') = len (a*x) by Def1;
  hence thesis by A1,A3,Th3;
end;
