reserve c for Complex;
reserve r for Real;
reserve m,n for Nat;
reserve f for complex-valued Function;
reserve f,g for differentiable Function of REAL,REAL;
reserve L for non empty ZeroStr;
reserve x for Element of L;

theorem Th30:
  for L being add-associative right_zeroed right_complementable
        non empty addLoopStr
  for x being Element of L holds
  -seq(n,x) = seq(n,-x)
  proof
    let L be add-associative right_zeroed right_complementable
    non empty addLoopStr;
    let x be Element of L;
    let a be Element of NAT;
    dom(-seq(n,x)) = NAT by FUNCT_2:def 1;
    then
A1: (-seq(n,x))/.a = -(seq(n,x))/.a by VFUNCT_1:def 5;
    per cases;
    suppose a = n;
      then seq(n,x).a = x & seq(n,-x).a = -x by Th24;
      hence thesis by A1;
    end;
    suppose a <> n;
      then seq(n,x).a = 0.L & seq(n,-x).a = 0.L by Th25;
      hence thesis by A1;
    end;
  end;
