:: deftheorem Def17 defines add2 GCD_1:def 17 :
for R being gcdDomain
for Amp being AmpleSet of R
for r1, r2, s1, s2 being Element of R st r1,r2 are_co-prime & s1,s2 are_co-prime & r2 = NF (r2,Amp) & s2 = NF (s2,Amp) holds
( ( r1 = 0. R implies add2 (r1,r2,s1,s2,Amp) = s2 ) & ( s1 = 0. R implies add2 (r1,r2,s1,s2,Amp) = r2 ) & ( gcd (r2,s2,Amp) = 1. R implies add2 (r1,r2,s1,s2,Amp) = r2 * s2 ) & ( (r1 * (s2 / (gcd (r2,s2,Amp)))) + (s1 * (r2 / (gcd (r2,s2,Amp)))) = 0. R implies add2 (r1,r2,s1,s2,Amp) = 1. R ) & ( not r1 = 0. R & not s1 = 0. R & not gcd (r2,s2,Amp) = 1. R & not (r1 * (s2 / (gcd (r2,s2,Amp)))) + (s1 * (r2 / (gcd (r2,s2,Amp)))) = 0. R implies add2 (r1,r2,s1,s2,Amp) = (r2 * (s2 / (gcd (r2,s2,Amp)))) / (gcd (((r1 * (s2 / (gcd (r2,s2,Amp)))) + (s1 * (r2 / (gcd (r2,s2,Amp))))),(gcd (r2,s2,Amp)),Amp)) ) );