reserve V for RealLinearSpace;
reserve p,q,u,v,w,y for VECTOR of V;
reserve a,b,c,d for Real;
reserve AS for non empty AffinStruct;
reserve a,b,c,d for Element of AS;
reserve x,z for object;

theorem Th26:
  (ex u,v st for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0
  ) implies OASpace(V) is OAffinSpace
proof
  assume
A1: ex u,v st for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0;
  then
A2: ( ex a,b,c,d being Element of OASpace(V) st not a,b // c,d & not a,b //
  d,c)& for a,b,c being Element of OASpace(V) ex d being Element of OASpace(V)
  st a,b // c,d & a,c // b,d & b<>d by Th23;
A3: for p,a,b,c being Element of OASpace(V) st p<>b & b,p // p,c ex d being
  Element of OASpace(V) st a,p // p,d & a,b // c,d by A1,Th23;
  ( ex a,b being Element of OASpace(V) st a<>b)& for a,b,c,d,p,q,r,s being
Element of OASpace(V) holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b
// p,q & a,b // r,s implies p,q // r,s) & (a, b // c,d implies b,a // d,c) & (a
  ,b // b,c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c,b)
  by A1,Th23;
  hence thesis by A2,A3,Def5,STRUCT_0:def 10;
end;
