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 Th47:
  f is dilatation & x,f.x,y are_collinear implies x,f.x,f.y are_collinear
proof
  assume
A1: f is dilatation;
  assume
A2: x,f.x,y are_collinear;
  now
    assume
A3: x<>y;
    x,f.x '||' x,y & x,y '||' f.x,f.y by A1,A2,Th34,DIRAF:def 5;
    then x,f.x '||' f.x,f.y by A3,DIRAF:23;
    then f.x,x '||' f.x,f.y by DIRAF:22;
    then f.x,x,f.y are_collinear by DIRAF:def 5;
    hence thesis by DIRAF:30;
  end;
  hence thesis by DIRAF:31;
end;
