reserve X for OrtAfPl;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1,c2,c3,d,d1,d2,d3,d4,e1,e2 for
  Element of X;
reserve a29,a39,b29,x9 for Element of the AffinStruct of X;
reserve A,K,M,N for Subset of X;
reserve A9,K9 for Subset of the AffinStruct of X;

theorem Th8:
  the AffinStruct of X is translational implies X is satisfying_des
proof
  assume
A1: the AffinStruct of X is translational;
  let a,a1,b,b1,c,c1;
  assume that
A2: not LIN a,a1,b and
A3: not LIN a,a1,c and
A4: a,a1 // b,b1 and
A5: a,a1 // c,c1 and
A6: a,b // a1,b1 and
A7: a,c // a1,c1;
  reconsider a9=a,a19=a1,b9=b,b19=b1,c9=c,c19=c1
    as Element of the AffinStruct of X;
  LIN a9,a19,a19 by AFF_1:7;
  then consider A9 be Subset of the AffinStruct of X such that
A8: A9 is being_line and
A9: a9 in A9 and
A10: a19 in A9 and
  a19 in A9 by AFF_1:21;
A11: a9,b9 // a19,b19 by A6,ANALMETR:36;
  b,b1 // a,a1 by A4,ANALMETR:59;
  then
A12: b9,b19 // a9,a19 by ANALMETR:36;
A13: a9<>a19
  proof
    assume a9=a19;
    then LIN a9,a19,b9 by AFF_1:7;
    hence contradiction by A2,ANALMETR:40;
  end;
  then for x9 holds x9 in A9 iff LIN a9,a19,x9 by A8,A9,A10,AFF_1:21,25;
  then A9=Line(a9,a19) by AFF_1:def 2;
  then
A14: b9,b19 // A9 by A13,A12,AFF_1:def 4;
A15: a9,c9 // a19,c19 by A7,ANALMETR:36;
  c,c1 // a,a1 by A5,ANALMETR:59;
  then
A16: c9,c19 // a9,a19 by ANALMETR:36;
A17: a9<>a19
  proof
    assume a9=a19;
    then LIN a9,a19,b9 by AFF_1:7;
    hence contradiction by A2,ANALMETR:40;
  end;
  then for x9 holds x9 in A9 iff LIN a9,a19,x9 by A8,A9,A10,AFF_1:21,25;
  then A9=Line(a9,a19) by AFF_1:def 2;
  then
A18: c9,c19 // A9 by A17,A16,AFF_1:def 4;
  LIN c9,c19,c19 by AFF_1:7;
  then consider N9 be Subset of the AffinStruct of X such that
A19: N9 is being_line and
A20: c9 in N9 and
A21: c19 in N9 and
  c19 in N9 by AFF_1:21;
  LIN b9,b19,b19 by AFF_1:7;
  then consider M9 be Subset of the AffinStruct of X such that
A22: M9 is being_line and
A23: b9 in M9 and
A24: b19 in M9 and
  b19 in M9 by AFF_1:21;
A25: A9<>M9 & A9<>N9
  proof
    assume A9=M9 or A9=N9;
    then LIN a9,a19,b9 or LIN a9,a19,c9 by A8,A9,A10,A23,A20,AFF_1:21;
    hence contradiction by A2,A3,ANALMETR:40;
  end;
A26: c9<>c19
  proof
    assume c9=c19;
    then c,a // c,a1 by A7,ANALMETR:59;
    then LIN c,a,a1 by ANALMETR:def 10;
    then LIN c9,a9,a19 by ANALMETR:40;
    then LIN a9,a19,c9 by AFF_1:6;
    hence contradiction by A3,ANALMETR:40;
  end;
  then for x9 holds x9 in N9 iff LIN c9,c19,x9 by A19,A20,A21,AFF_1:21,25;
  then N9=Line(c9,c19) by AFF_1:def 2;
  then
A27: A9 // N9 by A18,A26,AFF_1:def 5;
A28: b9<>b19
  proof
    assume b9=b19;
    then b,a // b,a1 by A6,ANALMETR:59;
    then LIN b,a,a1 by ANALMETR:def 10;
    then LIN b9,a9,a19 by ANALMETR:40;
    then LIN a9,a19,b9 by AFF_1:6;
    hence contradiction by A2,ANALMETR:40;
  end;
  then for x9 holds x9 in M9 iff LIN b9,b19,x9 by A22,A23,A24,AFF_1:21,25;
  then M9=Line(b9,b19) by AFF_1:def 2;
  then A9 // M9 by A14,A28,AFF_1:def 5;
  then b9,c9 // b19,c19 by A1,A8,A9,A10,A22,A23,A24,A19,A20,A21,A27,A25,A11,A15
;
  hence thesis by ANALMETR:36;
end;
