
theorem asymTT4:
for a,b,p,q,s be Real st
a > 0 & p > 0 & (s-b)/a < (s-q)/(-p) holds
(s-b)/a < (q-b)/(a+p) & (q-b)/(a+p) < (s-q)/(-p)
proof
 let a,b,p,q,s be Real;
 assume AA: a > 0;
 assume PP: p > 0;
 assume (s-b)/a < (s-q)/(-p);then
 (s-b)/a*a < (s-q)/(-p)*a by XREAL_1:68,AA; then
 a/a*(s-b) < (s-q)/(-p)*a by XCMPLX_1:75; then
 1*(s-b) < (s-q)/(-p)*a by XCMPLX_1:60,AA; then
 (s-b)*(-p) > (s-q)/(-p)*a*(-p) by XREAL_1:69,PP; then
 (s-b)*(-p) > (s-q)/(-p)*(-p)*a; then
 (-p)*s-b*(-p) > (s-q)*a by XCMPLX_1:87,PP;then
 (-p)*s-b*(-p)+p*s > s*a-q*a+p*s by XREAL_1:6;then
 -b*(-p)+a*q > s*a-q*a+p*s +a*q by XREAL_1:6;then
 (b*p+a*q)/(a+p) > s*(a+p)/(a+p) by XREAL_1:74,AA,PP; then
 A3:(b*p+a*q)/(a+p) > s by XCMPLX_1:89,AA,PP;
 (b*p+a*q)/(a+p)-b = ((b*p+a*q) - b*(a+p))/(a+p) by XCMPLX_1:126,AA,PP
 .=(a*(q - b))/(a+p); then
 (a*(q - b))/(a+p) > s-b by A3,XREAL_1:9; then
 a*(q - b)/(a+p)/a > (s-b)/a by XREAL_1:74,AA; then
 A4a:a*((q - b)/(a+p))/a > (s-b)/a by XCMPLX_1:74;
 (b*p+a*q)/(a+p)-q= ((b*p+a*q) - q*(a+p))/(a+p) by XCMPLX_1:126,AA,PP
 .=(-p)*(q-b)/(a+p); then
 (-p)*(q-b)/(a+p) > s-q by XREAL_1:9,A3; then
 (-p)*(q-b)/(a+p)/(-p) < (s-q)/(-p) by XREAL_1:75,PP;then
 (-p)*((q-b)/(a+p))/(-p) < (s-q)/(-p) by XCMPLX_1:74;
 hence thesis by PP,A4a,AA,XCMPLX_1:89;
end;
