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 Th9:
  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
A1: i1 * i2 = 1;
    then
A2: not i2 = 0;
A3: now
A4:   (- i1) * (- i2) = i1 * i2;
      assume that
A5:   i1 < 0 and
A6:   i2 < 0;
A7:   -i2 is Element of NAT by A6,Th3;
      -i1 is Element of NAT by A5,Th3;
      then - (- i1) = - 1 by A1,A7,A4,NAT_1:15;
      hence i1 = - 1 & i2 = - 1 by A1;
    end;
A8: now
      assume that
A9:   0 < i1 and
A10:  0 < i2;
A11:  i2 is Element of NAT by A10,Th3;
      i1 is Element of NAT by A9,Th3;
      hence i1 = 1 & i2 = 1 by A1,A11,NAT_1:15;
    end;
    not i1 = 0 by A1;
    hence thesis by A1,A2,A8,A3;
  end;
  thus thesis;
end;
