
theorem
  for a,b,c,d be positive Real holds a*b = c*d & a*c = b*d implies a = d
  proof
    let a,b,c,d be positive Real;
    assume a*b = c*d & a*c = b*d; then
    (a*b)*(a*c) = (c*d)*(b*d); then
    (a*a)*(b*c) = (d*d)*(b*c); then
    a*a = d*d by XCMPLX_1:5; then
    (a >= d & d >= a) by XREAL_1:96;
    hence thesis by XXREAL_0:1;
  end;
