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 Th49:
  f is dilatation implies ((f=id the carrier of OAS or for x holds
  f.x<>x) iff for x,y holds x,f.x '||' y,f.y )
proof
  assume
A1: f is dilatation;
A2: now
    assume
A3: for x,y holds x,f.x '||' y,f.y;
    assume f<>(id the carrier of OAS);
    then consider y such that
A4: f.y<>(id the carrier of OAS).y by FUNCT_2:63;
    given x such that
A5: f.x=x;
    x<>y by A5,A4;
    then consider z such that
A6: not x,y,z are_collinear by DIRAF:37;
    x,z '||' f.x,f.z by A1,Th34;
    then x,z,f.z are_collinear by A5,DIRAF:def 5;
    then
A7: z,f.z,x are_collinear by DIRAF:30;
    z,f.z,z are_collinear by DIRAF:31;
    then
A8: z,f.z '||' x,z by A7,DIRAF:34;
A9: f.y<>y by A4;
    x,y '||' f.x,f.y by A1,Th34;
    then
A10: x,y,f.y are_collinear by A5,DIRAF:def 5;
    then y,x,f.y are_collinear by DIRAF:30;
    then
A11: y,x '||' y,f.y by DIRAF:def 5;
A12: y,f.y,x are_collinear by A10,DIRAF:30;
A13: now
      assume z=f.z;
      then z,y '||' z,f.y by A1,Th34;
      then z,y,f.y are_collinear by DIRAF:def 5;
      then y,f.y,y are_collinear & y,f.y,z are_collinear by DIRAF:30,31;
      hence contradiction by A9,A12,A6,DIRAF:32;
    end;
    y,f.y '||' z,f.z by A3;
    then y,f.y '||' x,z by A13,A8,DIRAF:23;
    then y,x '||' x,z by A9,A11,DIRAF:23;
    then x,y '||' x,z by DIRAF:22;
    hence contradiction by A6,DIRAF:def 5;
  end;
  now
    assume
A14: f=(id the carrier of OAS) or for z holds f.z<>z;
    let x,y;
A15: x,y '||' f.x,f.y by A1,Th34;
A16: now
      assume
A17:  for z holds f.z<>z;
      assume
A18:  not x,f.x '||' y,f.y;
      then consider z such that
A19:  x,f.x,z are_collinear and
A20:  y,f.y,z are_collinear by A15,Th48;
      set t=f.z;
      x,f.x,t are_collinear by A1,A19,Th47;
      then
A21:  x,f.x '||' z,t by A19,DIRAF:34;
      y,f.y,t are_collinear by A1,A20,Th47;
      then
A22:  y,f.y '||' z,t by A20,DIRAF:34;
      z<>t by A17;
      hence contradiction by A18,A21,A22,DIRAF:23;
    end;
    f=(id the carrier of OAS) implies x,f.x '||' y,f.y by DIRAF:20;
    hence x,f.x '||' y,f.y by A14,A16;
  end;
  hence thesis by A2;
end;
