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
  (for a,b ex f st f is translation & f.a=b) implies AFP is translational
proof
  assume
A1: for a,b ex f st f is translation & f.a=b;
  now
    let a,a9,b,c,b9,c9;
    consider f such that
A2: f is translation and
A3: f.a=a9 by A1;
A4: f is dilatation by A2,TRANSGEO:def 11;
    assume ( not LIN a,a9,b)& not LIN a,a9,c & a,a9 // b,b9 & a,a9 // c,c9 &
    a,b // a9,b9 & a,c // a9,c9;
    then b9=f.b & c9=f.c by A2,A3,Th3;
    hence b,c // b9,c9 by A4,TRANSGEO:68;
  end;
  hence thesis by Th4;
end;
