
theorem SIL:
  for a,b be positive Real holds
  a + b > a * b iff 1/a + 1/b > 1
  proof
    let a,b be positive Real;
    A1: 1*a/(a*b) = 1/b & 1*b/(a*b) = 1/a by XCMPLX_1:91;
    A2: a + b > a*b implies 1/a + 1/b > 1
    proof
      assume a + b > a * b; then
      (a + b)/(a*b) > (a*b)/(a*b) by XREAL_1:74;
      hence thesis by XCMPLX_1:60,A1;
    end;
    1/a + 1/b > 1 implies a + b > a * b
    proof
      assume 1/a + 1/b > 1; then
      (a+b)/(a*b) > (a*b)/(a*b) by A1,XCMPLX_1:60;
      hence thesis by XREAL_1:72;
    end;
    hence thesis by A2;
  end;
