
theorem Th9:
  for OAP being OAffinSpace st OAP is satisfying_DES_1 holds
  Lambda(OAP) is Desarguesian
proof
  let OAP be OAffinSpace;
  set AP = Lambda(OAP);
  assume
A1: OAP is satisfying_DES_1;
  then
A2: OAP is satisfying_DES by Th6;
  for A,P,C being Subset of AP, o,a,b,c,a9,b9,c9 being Element of AP st o
in A & o in P & o in C & o<>a & o<>b & o<>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 AP;
    let o,a,b,c,a9,b9,c9 be Element of AP;
    reconsider o1=o,a1=a,b1=b,c1=c,a19=a9,b19=b9,c19=c9 as Element of OAP by
Th1;
    assume that
A3: o in A and
A4: o in P and
A5: o in C and
A6: o<>a and
A7: o<>b and
A8: o<>c and
A9: a in A and
A10: a9 in A and
A11: b in P and
A12: b9 in P and
A13: c in C and
A14: c9 in C and
A15: A is being_line and
A16: P is being_line and
A17: C is being_line and
A18: A<>P and
A19: A<>C and
A20: a,b // a9,b9 & a,c // a9,c9;
    LIN o,b,b9 by A4,A11,A12,A16,AFF_1:21;
    then
A21: o1,b1,b19 are_collinear by Th2;
A22: not o1,a1,b1 are_collinear & not o1,a1,c1 are_collinear
    proof
A23:  now
        assume LIN o,a,c;
        then consider X being Subset of Lambda(OAP) such that
A24:    X is being_line & o in X and
A25:    a in X and
A26:    c in X by AFF_1:21;
        X = C by A5,A8,A13,A17,A24,A26,AFF_1:18;
        hence contradiction by A3,A6,A9,A15,A19,A24,A25,AFF_1:18;
      end;
A27:  now
        assume LIN o,a,b;
        then consider X being Subset of Lambda(OAP) such that
A28:    X is being_line & o in X and
A29:    a in X and
A30:    b in X by AFF_1:21;
        X = P by A4,A7,A11,A16,A28,A30,AFF_1:18;
        hence contradiction by A3,A6,A9,A15,A18,A28,A29,AFF_1:18;
      end;
      assume not thesis;
      hence contradiction by A27,A23,Th2;
    end;
    LIN o,c,c9 by A5,A13,A14,A17,AFF_1:21;
    then
A31: o1,c1,c19 are_collinear by Th2;
A32: a1,b1 '||' a19,b19 & a1,c1 '||' a19,c19 by A20,DIRAF:38;
A33: now
      assume
A34:  a1,o1 // o1,a19;
      then
A35:  a1,b1 // b19,a19 & a1,c1 // c19,a19 by A21,A31,A22,A32,Th7;
      b1,o1 // o1,b19 & c1,o1 // o1,c19 by A21,A31,A22,A32,A34,Th7;
      then b1,c1 // c19,b19 by A1,A22,A34,A35;
      then b1,c1 '||' b19,c19 by DIRAF:def 4;
      hence thesis by DIRAF:38;
    end;
A36: now
      assume
A37:  o1,a1 // o1,a19;
      then
A38:  a1,b1 // a19,b19 & a1,c1 // a19,c19 by A21,A31,A22,A32,Th8;
      o1,b1 // o1,b19 & o1,c1 // o1,c19 by A21,A31,A22,A32,A37,Th8;
      then b1,c1 // b19,c19 by A2,A22,A37,A38;
      then b1,c1 '||' b19,c19 by DIRAF:def 4;
      hence thesis by DIRAF:38;
    end;
    LIN o,a,a9 by A3,A9,A10,A15,AFF_1:21;
    then o1,a1,a19 are_collinear by Th2;
    then Mid o1,a1,a19 or Mid a1,o1,a19 or Mid o1,a19,a1 by DIRAF:29;
    hence thesis by A33,A36,DIRAF:7,def 3;
  end;
  hence thesis by AFF_2:def 4;
end;
