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

theorem
  0 <= a & 0 <= c & a < b & c < d implies a*c < b*d
proof
  assume that
A1: 0<=a and
A2: 0<=c and
A3: a<b and
A4: c < d;
A5: a*c <= a*d by A1,A4,Lm12;
  a*d<b*d by A2,A3,A4,Lm13;
  hence thesis by A5,XXREAL_0:2;
end;
