reserve x,y for Element of REAL;
reserve i,j,k for Element of NAT;
reserve a,b for Element of REAL;

theorem
  for x,y being Element of REAL holds opp +(x,y) = +(opp x, opp y)
proof
  reconsider o = 0 as Element of REAL by Lm3;
  let x,y be Element of REAL;
  +(+(x,y),+(opp x, opp y)) = +(x,+(y,+(opp x, opp y))) by Th23
    .= +(x,+(opp x,+(y, opp y))) by Th23
    .= +(x,+(opp x,o)) by Def3
    .= +(x,opp x) by Th11
    .= 0 by Def3;
  hence thesis by Def3;
end;
