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

theorem Th11:
  p<>q & p,q // u,v & p,q // w,y implies u,v // w,y
proof
  assume that
A1: p<>q and
A2: p,q // u,v and
A3: p,q // w,y;
  now
    assume that
A4: u<>v and
A5: w<>y;
    consider a,b such that
A6: a*(q-p)=b*(v-u) and
A7: 0<a and
A8: 0<b by A1,A2,A4;
    0<a" by A7;
    then
A9: 0<a"*b by A8,XREAL_1:129;
    consider c,d such that
A10: c*(q-p)=d*(y-w) and
A11: 0<c and
A12: 0<d by A1,A3,A5;
A13: q-p=(c")*(d*(y-w)) by A10,A11,Th6
      .=(c"*d)*(y-w) by RLVECT_1:def 7;
    0<c" by A11;
    then
A14: 0<c"*d by A12,XREAL_1:129;
    q-p=(a")*(b*(v-u)) by A6,A7,Th6
      .=(a"*b)*(v-u) by RLVECT_1:def 7;
    hence thesis by A13,A9,A14;
  end;
  hence thesis;
end;
