
theorem
  for V being RealLinearSpace, OAS being OAffinSpace st OAS=OASpace(V)
holds ex u,v being VECTOR of V st for a,b being Real st a*u + b*v = 0.V holds a
  =0 & b=0
proof
  let V be RealLinearSpace, OAS be OAffinSpace such that
A1: OAS = OASpace(V);
  consider a,b,c,d being Element of OAS such that
A2: ( not a,b // c,d)& not a,b // d,c by ANALOAF:def 5;
  reconsider u=a,v=b,w=c,y=d as VECTOR of V by A1,Th3;
  take z1=v-u,z2=y-w;
  now
    let r1,r2 be Real;
    assume r1*z1+r2*z2 = 0.V;
    then
A3: r1*z1 = -(r2*z2) by RLVECT_1:6
      .= r2*(-z2) by RLVECT_1:25
      .= (-r2)*z2 by RLVECT_1:24;
    assume r1<>0 or r2<>0;
    then r1<>0 or -r2<>0;
    then u,v // w,y or u,v // y,w by A3,ANALMETR:14;
    hence r1=0 & r2=0 by A1,A2,GEOMTRAP:2;
  end;
  hence thesis;
end;
