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;

theorem Th41:
  Lambda(S) is AffinSpace
proof
  set AS=Lambda(S);
A1: ( 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 by Th39;
A2: 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 by Th39;
  ( 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 by Th39;
  hence thesis by A1,A2,Def6;
end;
