reserve V for RealLinearSpace;
reserve p,q,r,u,v,w,y,u1,v1,w1 for Element of V;
reserve a,b,c,d,a1,b1,c1,a2,b2,c2,a3,b3,e,f for Real;

theorem
  not p,q,r are_LinDep implies for u,v st u is not zero & v is not zero
  & not are_Prop u,v ex y st y is not zero & not u,v,y are_LinDep
proof
  assume
A1: not p,q,r are_LinDep;
  let u,v;
  assume u is not zero & v is not zero & not are_Prop u,v;
  then consider y such that
A2: not u,v,y are_LinDep by A1,Lm8;
  take y;
  thus y is not zero by A2,Th12;
  thus thesis by A2;
end;
