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 Th58:
  f is translation implies f is positive_dilatation
proof
  assume
A1: f is translation;
A2: now
    assume
A3: for x holds f.x<>x;
    for a,b holds a,b // f.a,f.b
    proof
      let a,b;
A4:   a<>f.a by A3;
      not a,f.a,b are_collinear implies a,b // f.a,f.b by A1,Lm5;
      hence thesis by A1,A4,Lm9;
    end;
    hence thesis by Th27;
  end;
  f=id the carrier of OAS implies thesis by Th28;
  hence thesis by A1,A2;
end;
