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

theorem Th98:
  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,Lm13;
  a*d<=b*d by A2,A3,A4,Lm12;
  hence thesis by A5,XXREAL_0:2;
end;
