reserve a,b,c,d for Real;
reserve r,s for Real;

theorem
  a < 0 & b <= a & c < 0 & d < c implies a*c < b*d
proof
  assume that
A1: 0 > a and
A2: a >= b and
A3: 0 > c and
A4: c > d;
A5: a*c < a*d by A1,A4,Lm24;
  a*d <= b*d by A2,A3,A4,Lm28;
  hence thesis by A5,XXREAL_0:2;
end;
