reserve x, y, z, r, s, t for Real;

theorem Th18:
  sgn (x * y) = sgn x * sgn y
proof
A1: 0 < x & 0 < y implies sgn (x * y) = sgn x * sgn y
  proof
    assume that
A2: 0 < x and
A3: 0 < y;
A4: sgn y = 1 by A3,Def2;
    0 * y < x * y & sgn x = 1 by A2,A3,Def2,XREAL_1:68;
    hence thesis by A4,Def2;
  end;
A5: 0 < x & y < 0 implies sgn (x * y) = sgn x * sgn y
  proof
    assume that
A6: 0 < x and
A7: y < 0;
    sgn y = -1 by A7,Def2;
    then
A8: sgn x * sgn y = 1 * (-1) by A6,Def2
      .= -1;
    x * y < 0 * y by A6,A7,XREAL_1:69;
    hence thesis by A8,Def2;
  end;
A9: x < 0 & y < 0 implies sgn (x * y) = sgn x * sgn y
  proof
    assume that
A10: x < 0 and
A11: y < 0;
    sgn y = -1 by A11,Def2;
    then
A12: sgn x * sgn y = (-1) * (-1) by A10,Def2
      .= 1;
    x * 0 < x * y by A10,A11,XREAL_1:69;
    hence thesis by A12,Def2;
  end;
A13: x < 0 & 0 < y implies sgn (x * y) = sgn x * sgn y
  proof
    assume that
A14: x < 0 and
A15: 0 < y;
    sgn y = 1 by A15,Def2;
    then
A16: sgn x * sgn y = -1 by A14,Def2;
    x * y < 0 * y by A14,A15,XREAL_1:68;
    hence thesis by A16,Def2;
  end;
  x = 0 or y = 0 implies sgn (x * y) = sgn x * sgn y
  proof
    assume
A17: x = 0 or y = 0;
    then sgn x = 0 or sgn y = 0 by Def2;
    hence thesis by A17;
  end;
  hence thesis by A1,A5,A13,A9;
end;
