Singular

Hi, if any of the more exprienced users could give me his just their gut feeling and/or advice, any help is greatly appreceated!

After trying for some time with different approaches namely the fractal walk and the hilbert driven BB, I am starting to wonder, wether it is reasonable to expect the following elimination problem to compute:
(dp groebner bases compute very nicely, dimension of the generated ideal is 20)

Eliminate y1,y2,y3,y4, to be able to express y5 in terms of the x(ik):

ring s = 0,(y1,y2,y4,y5,x11,x12,x13,x14,x21,x22,x23,x24,x31,x32,x33,x34,x41,x42,x43,x44,x51,x52,x53,x54), (dp);

poly f1 = y1^2+y2^2-y3^2+y4^2+y5^2+2*x13*y1*y4-2*x14*y1*y5+2*x14*y2*y4+2*x13*y2*y5-2*y1*x11-2*y2*x12-2*x11*x13*y4-2*x12*x14*y4+2*x11*x14*y5-2*x12*x13*y5+x11^2+x12^2;
poly f2 = y1^2+y2^2-y3^2+y4^2+y5^2+2*x23*y1*y4-2*x24*y1*y5+2*x24*y2*y4+2*x23*y2*y5-2*y1*x21-2*y2*x22-2*x21*x23*y4-2*x22*x24*y4+2*x21*x24*y5-2*x22*x23*y5+x21^2+x22^2;
poly f3 = y1^2+y2^2-y3^2+y4^2+y5^2+2*x33*y1*y4-2*x34*y1*y5+2*x34*y2*y4+2*x33*y2*y5-2*y1*x31-2*y2*x32-2*x31*x33*y4-2*x32*x34*y4+2*x31*x34*y5-2*x32*x33*y5+x31^2+x32^2;
poly f4 = y1^2+y2^2-y3^2+y4^2+y5^2+2*x43*y1*y4-2*x44*y1*y5+2*x44*y2*y4+2*x43*y2*y5-2*y1*x41-2*y2*x42-2*x41*x43*y4-2*x42*x44*y4+2*x41*x44*y5-2*x42*x43*y5+x41^2+x42^2;
poly f5 = y1^2+y2^2-y3^2+y4^2+y5^2+2*x53*y1*y4-2*x54*y1*y5+2*x54*y2*y4+2*x53*y2*y5-2*y1*x51-2*y2*x52-2*x51*x53*y4-2*x52*x54*y4+2*x51*x54*y5-2*x52*x53*y5+x51^2+x52^2;