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 Th12:
  for i being Nat, a being Complex holds (a*x).i = a*(x.i)
proof
  let i be Nat, a be Complex;
  reconsider aa = a as Element of COMPLEX by XCMPLX_0:def 2;
  per cases;
  suppose
A1: not i in dom (a*x); then
A2: not i in dom x by Th11;
    thus (a*x).i = a*0 by A1,FUNCT_1:def 2
      .= a*(x.i) by A2,FUNCT_1:def 2;
  end;
  suppose
A3: i in dom (a*x);
    set a9 = x.i;
A4: a*x = (multcomplex[;](aa,id COMPLEX))*x by Lm1; then
A5: a9 in dom(multcomplex[;](aa,id COMPLEX)) by A3,FUNCT_1:11;
    thus (a*x).i = (multcomplex[;](aa,id COMPLEX)).a9 by A3,A4,FUNCT_1:12
      .= multcomplex.(a,(id COMPLEX).a9) by A5,FUNCOP_1:32
      .= a*(x.i) by BINOP_2:def 5;
  end;
end;
