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 Th11:
  f is translation & g is translation & p,f.p // p,g.p implies p,f
  .p // p,(f*g).p
proof
  assume that
A1: f is translation and
A2: g is translation and
A3: p,f.p // p,g.p;
A4: f is dilatation by A1,TRANSGEO:def 11;
A5: now
    assume g<>id the carrier of AFP;
    then
A6: g.p<>p by A2,TRANSGEO:def 11;
    p,g.p // f.p,f.(g.p) by A4,TRANSGEO:68;
    then p,f.p // f.p,f.(g.p) by A3,A6,AFF_1:5;
    then f.p,p // f.p,f.(g.p) by AFF_1:4;
    then LIN f.p,p,f.(g.p) by AFF_1:def 1;
    then LIN p,f.p,f.(g.p) by AFF_1:6;
    then p,f.p // p,f.(g.p) by AFF_1:def 1;
    hence thesis by FUNCT_2:15;
  end;
  now
    assume g=id the carrier of AFP;
    then f*g=f by FUNCT_2:17;
    hence thesis by AFF_1:2;
  end;
  hence thesis by A5;
end;
