reserve r,t for Real;
reserve i for Integer;
reserve k,n for Nat;
reserve p for Polynomial of F_Real;
reserve e for Element of F_Real;
reserve L for non empty ZeroStr;
reserve z,z0,z1,z2 for Element of L;

theorem Th30:
  for L being add-associative right_zeroed right_complementable
        non empty addLoopStr
  for z0 being Element of L holds
  - <%z0%> = <%-z0%>
  proof
    let L be add-associative right_zeroed right_complementable
    non empty addLoopStr;
    let z0 be Element of L;
    set p = <%z0%>;
    set r = <%-z0%>;
    let n be Element of NAT;
A1: dom -p = NAT by FUNCT_2:def 1;
A2: (-p).n = (-p)/.n
    .= -(p/.n) by A1,VFUNCT_1:def 5
    .= -(p.n);
    per cases;
    suppose n = 0;
      then p.n = z0 & r.n = -z0 by POLYNOM5:32;
      hence thesis by A2;
    end;
    suppose n > 0;
      then n >= 0+1 by NAT_1:13;
      then p.n = 0.L & r.n = 0.L by POLYNOM5:32;
      hence thesis by A2;
    end;
  end;
