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
  not ex f st f is negative_dilatation & f is positive_dilatation
proof
  given f such that
A1: f is negative_dilatation & f is positive_dilatation;
  consider a,b such that
A2: a<>b by STRUCT_0:def 10;
  a,b // f.a,f.b & a,b // f.b,f.a by A1,Th27;
  then f.a,f.b // f.b,f.a by A2,ANALOAF:def 5;
  then f.a = f.b by ANALOAF:def 5;
  hence contradiction by A2,FUNCT_2:58;
end;
