reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;

theorem Th2:
  for p,q,p1,q1 being Element of OASpace(V) st p=u & q=v & p1=u1 &
  q1=v1 holds (p,q // p1,q1 iff u,v // u1,v1)
proof
A1: OASpace(V) = AffinStruct (#the carrier of V, DirPar(V)#) by ANALOAF:def 4;
  let p,q,p1,q1 be Element of OASpace(V) such that
A2: p=u & q=v & p1=u1 & q1=v1;
A3: now
    assume u,v // u1,v1;
    then [[p,q],[p1,q1]] in the CONGR of OASpace(V) by A2,A1,ANALOAF:22;
    hence p,q // p1,q1 by ANALOAF:def 2;
  end;
  now
    assume p,q // p1,q1;
    then [[u,v],[u1,v1]] in DirPar(V) by A2,A1,ANALOAF:def 2;
    hence u,v // u1,v1 by ANALOAF:22;
  end;
  hence thesis by A3;
end;
