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

theorem Th20:
  (ex u,v st (for a,b st a*u + b*v=0.V holds a=0 & b=0)) implies
  ex u,v,w,y st not u,v // w,y & not u,v // y,w
proof
  given u,v such that
A1: for a,b st a*u + b*v=0.V holds a=0 & b=0;
A2: u<>0.V & v<>0.V by A1,Th19;
A3: not 0.V,u // v,0.V
  proof
A4: now
      given a,b such that
A5:   0<a and
      0<b and
A6:   a*(u-0.V) = b*(0.V-v);
      a*u = a*(u-0.V) & b*(0.V-v)=b*(-v) by RLVECT_1:13,14;
      then a*u = -(b*v) by A6,RLVECT_1:25;
      then a*u + b*v = 0.V by RLVECT_1:5;
      hence contradiction by A1,A5;
    end;
    assume 0.V,u // v,0.V;
    hence contradiction by A2,A4;
  end;
  not 0.V,u // 0.V,v
  proof
A7: now
      given a,b such that
A8:   0<a and
      0<b and
A9:   a*(u-0.V) = b*(v-0.V);
      a*u = a*(u-0.V) & b*(v-0.V)=b*v by RLVECT_1:13;
      then 0.V = a*u - (b*v) by A9,RLVECT_1:15
        .= a*u + (b*(-v)) by RLVECT_1:25
        .= a*u + ((-b)*v) by RLVECT_1:24;
      hence contradiction by A1,A8;
    end;
    assume 0.V,u // 0.V,v;
    hence contradiction by A2,A7;
  end;
  hence thesis by A3;
end;
