reserve V for RealLinearSpace,
  o,p,q,r,s,u,v,w,y,y1,u1,v1,w1,u2,v2,w2 for Element of V,
  a,b,c,d,a1,b1,c1,d1,a2,b2,c2,d2,a3,b3,c3,d3 for Real,
  z for set;

theorem Th2:
  for u,v,u1,v1 holds ((for a,b,a1,b1 st a*u + b*v + a1*u1 + b1*v1
  = 0.V holds a=0 & b=0 & a1=0 & b1=0) implies u is not zero & v is not zero &
not are_Prop u,v & u1 is not zero & v1 is not zero & not are_Prop u1,v1 & not u
  ,v,u1 are_LinDep & not u1,v1,u are_LinDep)
proof
  let u,v,u1,v1;
  assume
A1: for a,b,a1,b1 st a*u + b*v + a1*u1 + b1*v1 = 0.V holds a=0 & b=0 &
  a1=0 & b1=0;
A2: now
    let d1,d2,d3;
    assume d1*u + d2*v + d3*u1 = 0.V;
    then 0.V = d1*u + d2*v + d3*u1 + 0.V by RLVECT_1:4
      .= d1*u + d2*v + d3*u1 + 0*v1 by RLVECT_1:10;
    hence d1=0 & d2=0 & d3=0 by A1;
  end;
  now
    let d1,d2,d3;
    assume d1*u1 + d2*v1 + d3*u = 0.V;
    then 0.V = d3*u + d1*u1 + d2*v1 by RLVECT_1:def 3
      .= d3*u + 0.V + d1*u1 + d2*v1 by RLVECT_1:4
      .= d3*u + 0*v + d1*u1 + d2*v1 by RLVECT_1:10;
    hence d1=0 & d2=0 & d3=0 by A1;
  end;
  hence thesis by A2,Th1;
end;
