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 Th54:
  f is translation & g is translation & f.a=g.a & not a,f.a,x are_collinear
  implies f.x=g.x
proof
  assume that
A1: f is translation and
A2: g is translation and
A3: f.a=g.a and
A4: not a,f.a,x are_collinear;
  set b=f.a, y=f.x, z=g.x;
A5: a,x '||' b,z & a,b '||' x,z by A2,A3,Th34,Th53;
 f is dilatation by A1;
  then
A6: a,x '||' b,y by Th34;
  a,b '||' x,y by A1,Th53;
  hence thesis by A4,A6,A5,PASCH:5;
end;
