
theorem asymTT3:
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) & (s-q)/(-p) = (q-b)/(a+p)
proof
 let a,b,p,q,s be Real;
 assume AA:a > 0;
 assume PP: p > 0;
 assume A4: (s-b)/a = (s-q)/(-p); then
 (s-b)*(-p) = (s-q)/(-p)*a*(-p) by XCMPLX_1:87,AA; then
 (s-b)*(-p) = (s-q)/(-p)*(-p)*a; then
 (-p)*s-b*(-p) = (s-q)*a by XCMPLX_1:87,PP; then
 (a*q + b*p)/(a+p) = s*(a+p)/(a+p); then
 s-b = (a*q + b*p)/(a+p)-b by XCMPLX_1:89,AA,PP
 .= (a*q + b*p)/(a+p)-b*((a+p)/(a+p)) by XCMPLX_1:88,AA,PP
 .= (a*q + b*p)/(a+p)-b*(a+p)/(a+p) by XCMPLX_1:74
 .= ( (a*q + b*p)-(b*a+b*p) )/(a+p) by XCMPLX_1:120
 .= a*(q-b)/(a+p); then
 (s-b)/a = a*((q-b)/(a+p))/a by XCMPLX_1:74
 .=(q-b)/(a+p) by XCMPLX_1:89,AA;
 hence thesis by A4;
end;
