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 Th12:
  u,v // u,u#v & u,v // u#v,v
proof
  set p = u#v;
  1*(v-u) = v-u by RLVECT_1:def 8
    .= 2*(p-u) by Th10;
  hence u,v // u,u#v by ANALOAF:def 1;
  1*(v-u) = v-u by RLVECT_1:def 8
    .= 2*(v-p) by Th10;
  hence thesis by ANALOAF:def 1;
end;
