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;
reserve AS for non empty AffinStruct;
reserve S for OAffinPlane;
reserve x,y,z,t,u for Element of S;

theorem
  AS is AffinPlane iff (ex x,y being Element of AS st x<>y) & (for x,y,z
,t,u,w being Element of AS holds x,y // y,x & x,y // z,z & (x<>y & x,y // z,t &
  x,y // u,w implies z,t // u,w) & (x,y // x,z implies y,x // y,z)) & (ex x,y,z
  being Element of AS st not x,y // x,z) & (for x,y,z being Element of AS ex t
being Element of AS st x,z // y,t & y<>t) & (for x,y,z being Element of AS ex t
being Element of AS st x,y // z,t & x,z // y,t) & (for x,y,z,t being Element of
AS st z,x // x,t & x<>z ex u being Element of AS st y,x // x,u & y,z // t,u) &
for x,y,z,t being Element of AS st not x,y // z,t ex u being Element of AS st x
  ,y // x,u & z,t // z,u by Def6,Def7,STRUCT_0:def 10;
