
theorem Th13:
  for V being RealLinearSpace, OAS being OAffinSpace st OAS =
  OASpace(V) holds Lambda(OAS) is Pappian
proof
  let V be RealLinearSpace, OAS be OAffinSpace such that
A1: OAS = OASpace(V);
  set AS = Lambda(OAS);
  for M,N being Subset of AS, o,a,b,c,a9,b9,c9 being Element
 of AS st M is being_line & N is being_line & M<>N & o in M & o in N & o
<>a & o<>a9 & o<>b & o<>b9 & o<>c & o<>c9 & a in M & b in M & c in M & a9 in N
  & b9 in N & c9 in N & a,b9 // b,a9 & b,c9 // c,b9 holds a,c9 // c,a9
  proof
    let M,N be Subset of AS, o,a,b,c,a9,b9,c9 be Element of AS
    such that
A2: M is being_line and
A3: N is being_line and
A4: M<>N and
A5: o in M and
A6: o in N and
A7: o<>a and
A8: o<>a9 and
A9: o<>b and
    o<>b9 and
A10: o<>c and
A11: o<>c9 and
A12: a in M and
A13: b in M and
A14: c in M and
A15: a9 in N and
A16: b9 in N and
A17: c9 in N and
A18: a,b9 // b,a9 and
A19: b,c9 // c,b9;
    reconsider o1=o,a1=a,b1=b,c1=c,a19=a9,b19=b9,c19=c9 as Element of OAS by
Th1;
    reconsider q=o1,u=a1,v=b1,w=c1,u9=a19,v9=b19,w9=c19 as VECTOR of V by A1
,Th3;
    b1,c19 '||' c1,b19 by A19,DIRAF:38;
    then
A20: v,w9 '||' w,v9 by A1,Th4;
A21: not q,v '||' q,w9 & not q,v '||' q,u9
    proof
      assume not thesis;
      then o1,b1 '||' o1,c19 or o1,b1 '||' o1,a19 by A1,Th4;
      then o,b // o,c9 or o,b // o,a9 by DIRAF:38;
      then LIN o,b,c9 or LIN o,b,a9 by AFF_1:def 1;
      then c9 in M or a9 in M by A2,A5,A9,A13,AFF_1:25;
      hence contradiction by A2,A3,A4,A5,A6,A8,A11,A15,A17,AFF_1:18;
    end;
    LIN o,c,b by A2,A5,A13,A14,AFF_1:21;
    then o,c // o,b by AFF_1:def 1;
    then o1,c1 '||' o1,b1 by DIRAF:38;
    then q,w '||' q,v by A1,Th4;
    then consider r2 being Real such that
A22: w-q = r2*(v-q) and
A23: r2<>0 by A9,A10,Lm2;
A24: -r2<>0 by A23;
    LIN o,a,b by A2,A5,A12,A13,AFF_1:21;
    then o,a // o,b by AFF_1:def 1;
    then o1,a1 '||' o1,b1 by DIRAF:38;
    then q,u '||' q,v by A1,Th4;
    then consider r1 being Real such that
A25: u-q = r1*(v-q) and
A26: r1<>0 by A7,A9,Lm2;
A27: (-r1)*(q-v) = r1*(-(q-v)) by RLVECT_1:24
      .= u-q by A25,RLVECT_1:33;
    LIN o,c9,b9 by A3,A6,A16,A17,AFF_1:21;
    then o,c9 // o,b9 by AFF_1:def 1;
    then o1,c19 '||' o1,b19 by DIRAF:38;
    then
A28: q,w9 '||' q,v9 by A1,Th4;
    (-r2)*(q-v) = r2*(-(q-v)) by RLVECT_1:24
      .= w-q by A22,RLVECT_1:33;
    then
A29: q-v = (-r2)"*(w-q) by A24,ANALOAF:5;
    (-r2)" <>0 by A24,XCMPLX_1:202;
    then v9 = q + (-((-r2)"))"*(w9-q) by A20,A29,A28,A21,Th10
      .= q + (-(-(r2")))"*(w9-q) by XCMPLX_1:222
      .= q+ r2*(w9-q);
    then
A30: v9-q = r2*(w9-q) by RLSUB_2:61;
    LIN o,a9,b9 by A3,A6,A15,A16,AFF_1:21;
    then o,a9 // o,b9 by AFF_1:def 1;
    then o1,a19 '||' o1,b19 by DIRAF:38;
    then
A31: q,u9 '||' q,v9 by A1,Th4;
    a1,b19 '||' b1,a19 by A18,DIRAF:38;
    then b1,a19 '||' a1,b19 by DIRAF:22;
    then
A32: v,u9 '||' u,v9 by A1,Th4;
    r1"<>0 by A26,XCMPLX_1:202;
    then
A33: r1"*r2<>0 by A23,XCMPLX_1:6;
    set s=r1*(r2");
A34: u-q = r1*(r2"*(w-q)) by A25,A22,A23,ANALOAF:6
      .= s*(w-q) by RLVECT_1:def 7;
    -r1<>0 by A26;
    then
A35: (-r1)" <>0 by XCMPLX_1:202;
    -r1<>0 by A26;
    then q-v = (-r1)"*(u-q) by A27,ANALOAF:6;
    then v9 = q + (-((-r1)"))"*(u9-q) by A32,A35,A31,A21,Th10
      .= q + (-(-(r1")))"*(u9-q) by XCMPLX_1:222
      .= q+ r1*(u9-q);
    then v9-q = r1*(u9-q) by RLSUB_2:61;
    then u9-q = r1"*(r2*(w9-q)) by A26,A30,ANALOAF:6
      .= (r1"*r2)*(w9-q) by RLVECT_1:def 7;
    then
A36: w9-q = (r1"*r2)"*(u9-q) by A33,ANALOAF:6
      .= ((r1")"*(r2"))*(u9-q) by XCMPLX_1:204
      .= s*(u9-q);
    1*(w9-u) = w9-u by RLVECT_1:def 8
      .= s*(u9-q) - s*(w-q) by A36,A34,Lm3
      .= s*((u9-q)-(w-q)) by RLVECT_1:34
      .= s*(u9-w) by Lm3;
    then u,w9 // w,u9 or u,w9 // u9,w by ANALMETR:14;
    then u,w9 '||' w,u9 by GEOMTRAP:def 1;
    then a1,c19 '||' c1,a19 by A1,Th4;
    hence thesis by DIRAF:38;
  end;
  hence thesis by AFF_2:def 2;
end;
