reserve V for RealLinearSpace;
reserve p,q,u,v,w,y for VECTOR of V;
reserve a,b,c,d for Real;

theorem Th13:
  u,v // v,w implies u,v // u,w
proof
  assume
A1: u,v // v,w;
  now
    assume u<>v & v<>w;
    then consider a,b such that
A2: a*(v-u)=b*(w-v) and
A3: 0<a and
A4: 0<b by A1;
A5: 0<a+b by A3,A4;
    b*(w-u)=b*((w-v)+(v-u)) by Th1
      .=a*(v-u)+b*(v-u) by A2,RLVECT_1:def 5
      .=(a+b)*(v-u) by RLVECT_1:def 6;
    hence thesis by A4,A5;
  end;
  hence thesis by Th8;
end;
