
theorem Th17:
  for V being RealLinearSpace, OAS being OAffinSpace st OAS =
  OASpace(V) holds Lambda(OAS) is translational
proof
  let V be RealLinearSpace, OAS be OAffinSpace such that
A1: OAS = OASpace(V);
  set AS = Lambda(OAS);
  for A,P,C being Subset of AS, a,b,c,a9,b9,c9 being Element
 of AS st A // P & A // C & a in A & a9 in A & b in P & b9 in P & c in C
& c9 in C & A is being_line & P is being_line & C is being_line & A<>P & A<>C &
  a,b // a9,b9 & a,c // a9,c9 holds b,c // b9,c9
  proof
    let A,P,C be Subset of AS, a,b,c,a9,b9,c9 be Element of AS
    such that
A2: A // P and
A3: A // C and
A4: a in A and
A5: a9 in A and
A6: b in P and
A7: b9 in P and
A8: c in C and
A9: c9 in C and
A10: A is being_line and
A11: P is being_line and
A12: C is being_line and
A13: A<>P and
A14: A<>C and
A15: a,b // a9,b9 and
A16: a,c // a9,c9;
    reconsider a1=a,b1=b,c1=c,a19=a9,b19=b9,c19=c9 as Element of OAS by Th1;
    reconsider u=a1,v=b1,w=c1,u9=a19 as VECTOR of V by A1,Th3;
A17: now
      assume
A18:  a<>a9;
A19:  not a1,a19,b1 are_collinear
      proof
        assume a1,a19,b1 are_collinear;
        then LIN a,a9,b by Th2;
        then b in A by A4,A5,A10,A18,AFF_1:25;
        hence contradiction by A2,A6,A13,AFF_1:45;
      end;
A20:  not a1,a19,c1 are_collinear
      proof
        assume a1,a19,c1 are_collinear;
        then LIN a,a9,c by Th2;
        then c in A by A4,A5,A10,A18,AFF_1:25;
        hence contradiction by A3,A8,A14,AFF_1:45;
      end;
      a,a9 // c,c9 by A3,A4,A5,A8,A9,AFF_1:39;
      then
A21:  a1,a19 '||' c1,c19 by DIRAF:38;
      a,a9 // b,b9 by A2,A4,A5,A6,A7,AFF_1:39;
      then
A22:  a1,a19 '||' b1,b19 by DIRAF:38;
      set v99= (u9+v)-u,w99=(u9+w)-u;
      reconsider b199=v99,c199=w99 as Element of OAS by A1,Th3;
      w99-v99 = (u9+w) - (((u9+v)-u) + u) by RLVECT_1:27
        .= (u9+w) - (u9+v) by RLSUB_2:61
        .= ((w+u9)-u9) - v by RLVECT_1:27
        .= w - v by RLSUB_2:61;
      then v,w // v99,w99 by ANALOAF:15;
      then
A23:  v,w '||' v99,w99 by GEOMTRAP:def 1;
      u,u9 // v,v99 by ANALOAF:16;
      then u,u9 '||' v,v99 by GEOMTRAP:def 1;
      then
A24:  a1,a19 '||' b1,b199 by A1,Th4;
      u,w // u9,w99 by ANALOAF:16;
      then u,w '||' u9,w99 by GEOMTRAP:def 1;
      then
A25:  a1,c1 '||' a19,c199 by A1,Th4;
      u,u9 // w,w99 by ANALOAF:16;
      then u,u9 '||' w,w99 by GEOMTRAP:def 1;
      then
A26:  a1,a19 '||' c1,c199 by A1,Th4;
      u,v // u9,v99 by ANALOAF:16;
      then u,v '||' u9,v99 by GEOMTRAP:def 1;
      then
A27:  a1,b1 '||' a19,b199 by A1,Th4;
      a1,c1 '||' a19,c19 by A16,DIRAF:38;
      then
A28:  c199=c19 by A20,A21,A26,A25,PASCH:5;
      a1,b1 '||' a19,b19 by A15,DIRAF:38;
      then b199=b19 by A19,A22,A24,A27,PASCH:5;
      then b1,c1 '||' b19,c19 by A1,A28,A23,Th4;
      hence thesis by DIRAF:38;
    end;
    now
      assume
A29:  a=a9;
A30:  c =c9
      proof
        LIN a,c,c9 by A16,A29,AFF_1:def 1;
        then
A31:    LIN c,c9,a by AFF_1:6;
        assume c <>c9;
        then a in C by A8,A9,A12,A31,AFF_1:25;
        hence contradiction by A3,A4,A14,AFF_1:45;
      end;
      b=b9
      proof
        LIN a,b,b9 by A15,A29,AFF_1:def 1;
        then
A32:    LIN b,b9,a by AFF_1:6;
        assume b<>b9;
        then a in P by A6,A7,A11,A32,AFF_1:25;
        hence contradiction by A2,A4,A13,AFF_1:45;
      end;
      hence thesis by A30,AFF_1:2;
    end;
    hence thesis by A17;
  end;
  hence thesis by AFF_2:def 11;
end;
