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 Th78:
  f is dilatation & f.a=a & f.b=b & a<>b implies f=id the carrier of AFS
proof
  assume that
A1: f is dilatation and
A2: f.a=a and
A3: f.b=b and
A4: a<>b;
  now
    let x;
A5: now
      assume
A6:   LIN a,b,x;
      now
        consider z such that
A7:     not LIN a,b,z by A4,AFF_1:13;
        assume
A8:     x<>a;
A9:     not LIN a,z,x
        proof
          assume LIN a,z,x;
          then
A10:      LIN a,x,z by AFF_1:6;
          LIN a,x,a & LIN a,x,b by A6,AFF_1:6,7;
          hence contradiction by A8,A7,A10,AFF_1:8;
        end;
        f.z=z by A1,A2,A3,A7,Th77;
        hence f.x=x by A1,A2,A9,Th77;
      end;
      hence f.x=x by A2;
    end;
    not LIN a,b,x implies f.x=x by A1,A2,A3,Th77;
    hence f.x=(id the carrier of AFS).x by A5;
  end;
  hence thesis by FUNCT_2:63;
end;
