
theorem Th8:
  for f being complex-valued Function holds dom -f = dom f & for c
  being object holds (-f).c = -(f.c)
proof
  let f be complex-valued Function;
  thus
A1: dom -f = dom f by Def5;
  let c be object;
  per cases;
  suppose
    c in dom f;
    hence (-f).c = (-1)*f.c by A1,Def5
      .= -(f.c);
  end;
  suppose
A2: not c in dom f;
    hence (-f).c = -(0 qua Complex) by A1,FUNCT_1:def 2
      .= -(f.c) by A2,FUNCT_1:def 2;
  end;
end;
