RingMap TriaSystem -- apply ring map to a triangular system

i1 : R = QQ[a,b,c,d,e,f,g,h, MonomialOrder=>Lex];

i2 : F = {a*d - b*c, c*f - d*e, e*h - f*g};

i3 : T = triaSystem(R,F)

o3 = {a*d - b*c, c*f - d*e, e*h - f*g} / {d, f, h}

o3 : TriaSystem

i4 : S = QQ[x_0..x_7, MonomialOrder=>Lex];

i5 : f = map(S,R,gens S)

o5 = map (S, R, {x , x , x , x , x , x , x , x })
                  0   1   2   3   4   5   6   7

o5 : RingMap S <-- R

i6 : f T

o6 = {x x  - x x , x x  - x x , x x  - x x } / {x , x , x }
       0 3    1 2   2 5    3 4   4 7    5 6      3   5   7

o6 : TriaSystem