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 Th50:
  f is dilatation & f.a=a & f.b=b & not a,b,x are_collinear implies f.x=x
proof
  assume that
A1: f is dilatation and
A2: f.a=a and
A3: f.b=b and
A4: not a,b,x are_collinear;
  a,x '||' a,f.x by A1,A2,Th34;
  then a,x,f.x are_collinear by DIRAF:def 5;
  then
A5: x,f.x,a are_collinear by DIRAF:30;
  b,x '||' b,f.x by A1,A3,Th34;
  then b,x,f.x are_collinear by DIRAF:def 5;
  then
A6: x,f.x,x are_collinear & x,f.x,b are_collinear by DIRAF:30,31;
  assume f.x<>x;
  hence contradiction by A4,A5,A6,DIRAF:32;
end;
