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 Th9:
  f is translation & g is translation & not LIN a,f.a,g.a implies f*g=g*f
proof
  assume that
A1: f is translation and
A2: g is translation;
A3: g is dilatation by A2,TRANSGEO:def 11;
  then
A4: a,f.a // g.a,g.(f.a) by TRANSGEO:68;
  assume
A5: not LIN a,f.a,g.a;
  a,g.a // f.a,g.(f.a) by A2,A3,TRANSGEO:82;
  then f.(g.a)=g.(f.a) by A1,A5,A4,Th3;
  then (f*g).a=g.(f.a) by FUNCT_2:15;
  then
A6: (f*g).a=(g*f).a by FUNCT_2:15;
  f*g is translation & g*f is translation by A1,A2,TRANSGEO:86;
  hence thesis by A6,TRANSGEO:84;
end;
