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 Th13:
  AFP is Fanoian & f is translation & f*f=id the carrier of AFP
  implies f=id the carrier of AFP
proof
  set a = the Element of AFP;
  assume that
A1: AFP is Fanoian and
A2: f is translation and
A3: f*f=id the carrier of AFP;
  assume f<>id the carrier of AFP;
  then a<>f.a by A2,TRANSGEO:def 11;
  then consider b such that
A4: not LIN a,f.a,b by AFF_1:13;
A5: f is dilatation by A2,TRANSGEO:def 11;
  then
A6: a,b // f.a,f.b by TRANSGEO:68;
  f.b,a // f.(f.b),f.a by A5,TRANSGEO:68;
  then f.b,a // (f*f).b,f.a by FUNCT_2:15;
  then f.b,a // b,f.a by A3;
  then
A7: a,f.b // f.a,b by AFF_1:4;
  a,f.a // b,f.b by A2,A5,TRANSGEO:82;
  hence contradiction by A1,A4,A6,A7;
end;
