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 Th61:
  f is dilatation & f.p=p & q<>p & Mid q,p,f.q & not p,x,y are_collinear
  implies x,y // f.y,f.x
proof
  assume
A1: f is dilatation;
  assume
A2: f.p=p & q<>p & Mid q,p,f.q;
  then Mid x,p,f.x by A1,Th60;
  then
A3: x,p // p,f.x by DIRAF:def 3;
  Mid y,p,f.y by A1,A2,Th60;
  then
A4: y,p // p,f.y by DIRAF:def 3;
  x,y '||' f.x,f.y by A1,Th34;
  hence thesis by A3,A4,PASCH:12;
end;
