
theorem Th10:
  for A being set, f being Function of A, COMPLEX holds
    support f = support -f
proof
  let A be set, f be Function of A, COMPLEX;
  for x being object holds x in support f iff x in support -f
  proof
    let x be object;
    hereby
      assume A1: x in support f;
      then A2: x in dom f;
      then A3: x in dom -f by CFUNCT_1:5;
      reconsider A as non empty set by A1;
      reconsider y = x as Element of A by A2;
      (-f).x = (-f)/.y by A3, PARTFUN1:def 6
        .= -(f/.y) by CFUNCT_1:66
        .= -(f.x) by A1, PARTFUN1:def 6;
      then (-f).x <> 0 by A1, PRE_POLY:def 7;
      hence x in support -f by PRE_POLY:def 7;
    end;
    assume A4: x in support -f;
      then A5: x in dom -f;
      then A6: x in dom f by CFUNCT_1:5;
      reconsider A as non empty set by A4;
      reconsider y = x as Element of A by A5;
      (-f).x = (-f)/.y by A4, PARTFUN1:def 6
        .= -(f/.y) by CFUNCT_1:66
        .= -(f.x) by A6, PARTFUN1:def 6;
      then f.x <> 0 by A4, PRE_POLY:def 7;
      hence x in support f by PRE_POLY:def 7;
  end;
  hence thesis by TARSKI:2;
end;
