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

theorem Th14:
  u,v // u,w implies u,v // v,w or u,w // w,v
proof
  assume
A1: u,v // u,w;
  now
    assume u<>v & u<>w;
    then consider a,b such that
A2: a*(v-u)=b*(w-u) and
A3: 0<a and
A4: 0<b by A1;
    w-v=(w-u)+(u-v) by Th1
      .=(w-u)-(v-u) by RLVECT_1:33;
    then
A5: a*(w-v)=a*(w-u)-b*(w-u) by A2,RLVECT_1:34
      .=(a-b)*(w-u) by RLVECT_1:35
      .=(b-a)*(u-w) by Th4;
    v-w=(v-u)+(u-w) by Th1
      .=(v-u)-(w-u) by RLVECT_1:33;
    then
A6: b*(v-w)=b*(v-u)-a*(v-u) by A2,RLVECT_1:34
      .=(b-a)*(v-u) by RLVECT_1:35
      .=(a-b)*(u-v) by Th4;
A7: now
      assume a<>b;
      then 0<a-b or 0<b-a by XREAL_1:55;
      then v,u // w,v or w,u // v,w by A3,A4,A6,A5;
      hence thesis by Th12;
    end;
    now
      assume a=b;
      then (-u)+v= (-u)+w by A2,A3,RLVECT_1:36;
      then v=w by RLVECT_1:8;
      hence thesis;
    end;
    hence thesis by A7;
  end;
  hence thesis;
end;
