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 Th4:
  for p,q,p1,q1 being Element of (the carrier of AMSpace(V,w,y)) st
  p=u & q=v & p1=u1 & q1=v1 holds (p,q // p1,q1 iff u,v '||' u1,v1)
proof
  let p,q,p1,q1 be Element of (the carrier of AMSpace(V,w,y)) such that
A1: p=u & q=v & p1=u1 & q1=v1;
A2: now
    assume p,q // p1,q1;
    then ex a,b st a*(v-u) = b*(v1-u1) & (a<>0 or b<>0) by A1,ANALMETR:22;
    then u,v // u1,v1 or u,v // v1,u1 by ANALMETR:14;
    hence u,v '||' u1,v1;
  end;
  now
    assume u,v '||' u1,v1;
    then u,v // u1,v1 or u,v // v1,u1;
    then ex a,b st a*(v-u) = b*(v1-u1) & (a<>0 or b<>0) by ANALMETR:14;
    hence p,q // p1,q1 by A1,ANALMETR:22;
  end;
  hence thesis by A2;
end;
