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 Th1:
  ex a,b,c st LIN a,b,c & a<>b & b<>c & c <>a
proof
  consider a,p being Element of X such that
A1: a<>p by ANALMETR:39;
  consider b such that
A2: a,p _|_ p,b and
A3: p<>b by ANALMETR:39;
  reconsider a9=a,b9=b,p9=p as Element of the AffinStruct of X;
  consider c such that
A4: p,c _|_ a,b and
A5: LIN a,b,c by ANALMETR:69;
  take a,b,c;
  thus LIN a,b,c by A5;
  thus a<>b
  proof
    assume not thesis;
    then a,p _|_ a,p by A2,ANALMETR:61;
    hence contradiction by A1,ANALMETR:39;
  end;
  thus b<>c
  proof
    assume not thesis;
    then a,p // a,b by A2,A3,A4,ANALMETR:63;
    then LIN a,p,b by ANALMETR:def 10;
    then LIN a9,p9,b9 by ANALMETR:40;
    then LIN p9,a9,b9 by AFF_1:6;
    then LIN p,a,b by ANALMETR:40;
    then p,a // p,b by ANALMETR:def 10;
    then a,p _|_ p,a by A2,A3,ANALMETR:62;
    then a,p _|_ a,p by ANALMETR:61;
    hence contradiction by A1,ANALMETR:39;
  end;
  assume not thesis;
  then a,p _|_ a,b by A4,ANALMETR:61;
  then p,b // a,b by A1,A2,ANALMETR:63;
  then b,p // b,a by ANALMETR:59;
  then LIN b,p,a by ANALMETR:def 10;
  then LIN b9,p9,a9 by ANALMETR:40;
  then LIN p9,a9,b9 by AFF_1:6;
  then LIN p,a,b by ANALMETR:40;
  then p,a // p,b by ANALMETR:def 10;
  then a,p _|_ p,a by A2,A3,ANALMETR:62;
  then a,p _|_ a,p by ANALMETR:61;
  hence contradiction by A1,ANALMETR:39;
end;
