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 Th59:
  f is dilatation & f.p=p & Mid q,p,f.q & not p,q,x are_collinear
     implies Mid x,p,f.x
proof
  assume that
A1: f is dilatation and
A2: f.p=p and
A3: Mid q,p,f.q & not p,q,x are_collinear;
  q,x '||' f.q,f.x by A1,Th34;
  then
A4: q,x '||' f.x,f.q by DIRAF:22;
  p,x '||' p,f.x by A1,A2,Th34;
  then p,x,f.x are_collinear by DIRAF:def 5;
  hence thesis by A3,A4,PASCH:6;
end;
