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 Th10:
  (u = 0.V or v = 0.V or w = 0.V) implies u,v,w are_LinDep
proof
A1: for u,v,w st u=0.V holds u,v,w are_LinDep
  proof
    let u,v,w such that
A2: u=0.V;
    0.V = 0.V + 0.V
      .= 1*u + 0.V by A2
      .= 1*u + 0 * v by RLVECT_1:10
      .= 1*u + 0*v + 0.V
      .= 1*u + 0*v + 0*w by RLVECT_1:10;
    hence thesis;
  end;
A3: now
    assume v=0.V;
    then v,w,u are_LinDep by A1;
    hence thesis by Th5;
  end;
A4: now
    assume w=0.V;
    then w,u,v are_LinDep by A1;
    hence thesis by Th5;
  end;
  assume u=0.V or v=0.V or w=0.V;
  hence thesis by A1,A3,A4;
end;
