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
  f is dilatation implies f is positive_dilatation or f is negative_dilatation
proof
  assume
A1: f is dilatation;
A2: now
    given p such that
A3: f.p=p;
A4: now
      given q such that
A5:   not p,q // p,f.q;
      p,q '||' p,f.q by A1,A3,Th34;
      then
A6:   p,q // f.q,p by A5,DIRAF:def 4;
      then q,p // p,f.q by DIRAF:2;
      then
A7:   Mid q,p,f.q by DIRAF:def 3;
      p<>q by A5,A6,DIRAF:2;
      hence f is negative_dilatation by A1,A3,A7,Th63;
    end;
    now
      assume for x holds p,x // p,f.x;
      then for x,y holds x,y // f.x,f.y by A1,A3,Th64;
      hence f is positive_dilatation by Th27;
    end;
    hence thesis by A4;
  end;
  now
    assume for x holds f.x<>x;
    then f is translation by A1;
    hence f is positive_dilatation by Th58;
  end;
  hence thesis by A2;
end;
