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 Th9:
  for x being FinSequence of COMPLEX holds (-x).i = -x.i
proof
  let x be FinSequence of COMPLEX;
  per cases;
  suppose
A1: not i in dom -x; then
A2: not i in dom x by VALUED_1:8;
    thus (-x).i = -0 by A1,FUNCT_1:def 2
      .= -x.i by A2,FUNCT_1:def 2;
  end;
  suppose
A3: i in dom -x;
    set r = x.i;
    -x = compcomplex*x by SEQ_4:def 8;
    then (-x).i = compcomplex.r by A3,FUNCT_1:12;
    hence thesis by BINOP_2:def 1;
  end;
end;
