reserve A for non empty set,
  a,b,x,y,z,t for Element of A,
  f,g,h for Permutation of A;
reserve R for Relation of [:A,A:];
reserve AS for non empty AffinStruct;
reserve a,b,x,y for Element of AS;
reserve CS for CongrSpace;
reserve OAS for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u for Element of OAS;
reserve f,g for Permutation of the carrier of OAS;

theorem
  f is translation & g is translation implies (f*g) is translation
proof
  assume that
A1: f is translation and
A2: g is translation;
  f is dilatation & g is dilatation by A1,A2;
  then
A3: (f*g) is dilatation by Th44;
  now
    assume
A4: (f*g)<>(id the carrier of OAS);
    for x holds (f*g).x<>x
    proof
      given x such that
A5:   (f*g).x=x;
      f.(g.x)=x by A5,FUNCT_2:15;
      then
A6:   g.x=f".x by Th2;
      f" is translation by A1,Th56;
      then f*g=f*f" by A2,A6,Th55;
      hence contradiction by A4,FUNCT_2:61;
    end;
    hence thesis by A3;
  end;
  hence thesis by A3;
end;
