reserve MS for OrtAfPl;
reserve MP for OrtAfSp;

theorem Th6:
  for a,b,c,d1,d2 being Element of MS st not LIN a,b,c & d1,a _|_ b
  ,c & d1,b _|_ a,c & d2,a _|_ b,c & d2,b _|_ a,c holds d1=d2
proof
  let a,b,c,d1,d2 be Element of MS such that
A1: not LIN a,b,c and
A2: d1,a _|_ b,c and
A3: d1,b _|_ a,c and
A4: d2,a _|_ b,c and
A5: d2,b _|_ a,c;
  reconsider a9=a,b9=b,c9=c,d19=d1,d29=d2 as Element of the AffinStruct of MS;
  assume
A6: d1<>d2;
  b<>c by A1,Th1;
  then d1,a // d2,a by A2,A4,ANALMETR:63;
  then d19,a9 // d29,a9 by ANALMETR:36;
  then a9,d19 // a9, d29 by AFF_1:4;
  then LIN a9,d19,d29 by AFF_1:def 1;
  then
A7: LIN d19,d29,a9 by AFF_1:6;
  a<>c by A1,Th1;
  then d1,b // d2,b by A3,A5,ANALMETR:63;
  then d19,b9 // d29,b9 by ANALMETR:36;
  then b9,d19 // b9,d29 by AFF_1:4;
  then LIN b9,d19,d29 by AFF_1:def 1;
  then
A8: LIN d19,d29,b9 by AFF_1:6;
  set X9=Line(a9,b9);
  reconsider X=X9 as Subset of MS;
A9: b<>a by A1,Th1;
  then
A10: X9 is being_line by AFF_1:def 3;
  then
A11: X is being_line by ANALMETR:43;
A12: a9 in X9 by AFF_1:15;
A13: b9 in X9 by AFF_1:15;
  LIN d19,d29,d29 by AFF_1:7;
  then
A14: d2 in X by A6,A9,A7,A8,A10,A12,A13,AFF_1:8,25;
  LIN d19,d29,d19 by AFF_1:7;
  then
A15: d1 in X by A6,A9,A7,A8,A10,A12,A13,AFF_1:8,25;
  a<>d1 or a<>d2 by A6;
  then
A16: b,c _|_ X by A2,A4,A12,A15,A14,A11,ANALMETR:55;
  b<>d1 or b<>d2 by A6;
  then a,c _|_ X by A3,A5,A13,A15,A14,A11,ANALMETR:55;
  then a,c // b,c by A16,Th2;
  then a9,c9 // b9,c9 by ANALMETR:36;
  then c9,b9 // c9,a9 by AFF_1:4;
  then LIN c9,b9,a9 by AFF_1:def 1;
  then LIN a9,b9,c9 by AFF_1:6;
  hence contradiction by A1,ANALMETR:40;
end;
