
theorem asymTT2:
for a,b,p,q be Real st
a > 0 & p > 0 & (-b)/a < q/p
holds (-b)/a < (q-b)/(a+p) & (q-b)/(a+p) < q/p & (a*q+b*p)/(a+p) > 0
proof
 let a,b,p,q be Real;
 assume A1:a > 0;
 assume A2: p > 0;
 assume (-b)/a < q/p; then
A4a: (-b)/a -(-b)/a < q/p -(-b)/a by XREAL_1:9;
 q/p - (-b)/a = (q*a-(-b)*p)/(p*a) by A1,A2,XCMPLX_1:130
 .=(q*a+b*p)/(p*a); then
 A5: a*q+b*p>0 by A4a,A1,A2;
 (q-b)/(a+p)-((-b)/a)=
 (((q-b) * a) - ((-b) *(a+p))) / ((a+p) * a) by XCMPLX_1:130,A1,A2::A12,
 .=(a*q+b*p)/((a+p)*a); then
 A6: (q-b)/(a+p)-((-b)/a)+((-b)/a) > 0+((-b)/a) by XREAL_1:6,A2,A1,A5;
 (q-b)/(a+p) - q/p
  = (((q-b) * p) - (q *(a+p))) / ((a+p) * p) by A1,XCMPLX_1:130,A2
 .=(-(a*q+b*p))/((a+p)*p);
 then
 (q-b)/(a+p)-q/p + q/p < 0+q/p by XREAL_1:6,A1,A2,A5;
 hence thesis by A6,A5,A1,A2;
end;
