
theorem Th23:
  for A being Subset of REAL, x being Real holds -x ++ (x ++ A) = A
proof
  let A be Subset of REAL, x be Real;
  thus -x ++ (x ++ A) c= A
  proof
    let y be object;
    assume
A1: y in -x ++ (x ++ A);
    then reconsider y as Real;
    consider z being Real such that
A2: z in x ++ A and
A3: y = -x + z by A1,Lm1;
    ex t being Real st t in A & z = x + t by A2,Lm1;
    hence thesis by A3;
  end;
  let y be object;
  assume
A4: y in A;
  then reconsider y as Real;
  reconsider t = y - -x as Real;
  reconsider z = t -x as Real;
A5: z = -x + t;
  t in x ++ A by A4,Lm1;
  hence thesis by A5,Lm1;
end;
