per cases ( x = 0 or y = 0 or ( x > 0 & y > 0 ) ) ;

suppose ( x = 0 or y = 0 ) ; :: thesis: not x * y is negative

hence x * y >= 0 by Lm17; :: according to XXREAL_0:def 7 :: thesis: verum

end;

suppose ( x > 0 & y > 0 ) ; :: thesis: not x * y is negative

hence x * y >= 0 ; :: according to XXREAL_0:def 7 :: thesis: verum

end;

end;