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;
reserve AFS for AffinSpace;
reserve a,b,c,d,d1,d2,p,x,y,z,t for Element of AFS;
reserve f,g for Permutation of the carrier of AFS;

theorem
  f is dilatation & g is dilatation & f.a=g.a & f.b=g.b implies a=b or f =g
proof
  assume that
A1: f is dilatation and
A2: g is dilatation and
A3: f.a=g.a and
A4: f.b=g.b;
A5: (g"*f).b=g".(g.b) by A4,FUNCT_2:15
    .=b by Th2;
A6: g" is dilatation by A2,Th70;
  assume
A7: a<>b;
  (g"*f).a=g".(g.a) by A3,FUNCT_2:15
    .=a by Th2;
  then
A8: g"*f=(id the carrier of AFS) by A1,A7,A5,A6,Th71,Th78;
  now
    let x;
    g".(f.x)=(g"*f).x by FUNCT_2:15;
    then g".(f.x)=x by A8;
    hence g.x=f.x by Th2;
  end;
  hence thesis by FUNCT_2:63;
end;
