reserve x,y for set;
reserve X for non empty set;
reserve a,b,c,d for Element of X;
reserve S for OAffinSpace;
reserve a,b,c,d,p,q,r,x,y,z,t,u,w for Element of S;

theorem Th24:
  ex x,y,z st not x,y '||' x,z
proof
  consider x,y,z,t such that
A1: not x,y // z,t and
A2: not x,y // t,z by ANALOAF:def 5;
A3: x<>y by A1,Th4;
  now
    assume
A4: x,y '||' x,z;
    thus not x,y '||' x,t
    proof
      assume
A5:   x,y '||' x,t;
      then x,z '||' x,t by A3,A4,Th23;
      then z,x '||' z,t by Th21;
      then x,z '||' z,t by Th22;
      then x,y '||' z,t or x=z by A4,Th23;
      hence contradiction by A1,A2,A5;
    end;
  end;
  hence thesis;
end;
