reserve AS for AffinSpace,
  a,b,c,d,p,q,r,s,x for Element of AS;
reserve AFP for AffinPlane,
  a,a9,b,b9,c,c9,d,p,p9,q,q9,r,x,x9,y,y9,z for Element of AFP,
  A,C,P for Subset of AFP,
  f,g,h,f1,f2 for Permutation of the carrier of AFP;

theorem Th3:
  f is translation & not LIN a,f.a,x & a,f.a // x,y & a,x // f.a,y
  implies y=f.x
proof
  assume
A1: f is translation;
  then
A2: f is dilatation by TRANSGEO:def 11;
  then
A3: a,x // f.a,f.x by TRANSGEO:68;
  assume
A4: ( not LIN a,f.a,x)& a,f.a // x,y & a,x // f.a,y;
  a,f.a // x,f.x by A1,A2,TRANSGEO:82;
  hence thesis by A4,A3,TRANSGEO:80;
end;
