reserve X for set, x,y,z for object,
  k,l,n for Nat,
  r for Real;
reserve i,i0,i1,i2,i3,i4,i5,i8,i9,j for Integer;

theorem
  i1 * i2 = - 1 iff i1 = - 1 & i2 = 1 or i1 = 1 & i2 = - 1
proof
  thus i1 * i2 = - 1 implies i1 = - 1 & i2 = 1 or i1 = 1 & i2 = - 1
  proof
    assume i1 * i2 = - 1;
    then (- i1) * i2 = 1;
    then - (- i1) = - 1 & i2 = 1 or - i1 = - 1 & i2 = - 1 by Th9;
    hence thesis;
  end;
  assume i1 = - 1 & i2 = 1 or i1 = 1 & i2 = - 1;
  hence thesis;
end;
