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 Th31:
  for L being add-associative right_zeroed right_complementable
        non empty addLoopStr
  for z0,z1 being Element of L holds
  - <%z0,z1%> = <%-z0,-z1%>
  proof
    let L be add-associative right_zeroed right_complementable
    non empty addLoopStr;
    let z0,z1 be Element of L;
    set p = <%z0,z1%>;
    set r = <%-z0,-z1%>;
    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);
    (n = 0 or ... or n = 1) or n > 1;
    then per cases;
    suppose n = 0;
      then p.n = z0 & r.n = -z0 by POLYNOM5:38;
      hence thesis by A2;
    end;
    suppose n = 1;
      then p.n = z1 & r.n = -z1 by POLYNOM5:38;
      hence thesis by A2;
    end;
    suppose n > 1;
      then n >= 1+1 by NAT_1:13;
      then p.n = 0.L & r.n = 0.L by POLYNOM5:38;
      hence thesis by A2;
    end;
  end;
