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

theorem Th19:
  (for a,b st a*u + b*v=0.V holds a=0 & b=0) implies u<>v & u<>0.V & v<>0.V
proof
  assume
A1: for a,b st a*u + b*v=0.V holds a=0 & b=0;
  thus u<>v
  proof
    assume u=v;
    then u - v = 0.V by RLVECT_1:15;
    then 1*u + (-v) = 0.V by RLVECT_1:def 8;
    then 1*u + ((-jj)*v) = 0.V by RLVECT_1:16;
    hence contradiction by A1;
  end;
  thus u<>0.V
  proof
    assume u=0.V;
    then 1*u = 0.V by RLVECT_1:10;
    then 1*u + 0.V = 0.V by RLVECT_1:4;
    then jj*u + 0*v =0.V by RLVECT_1:10;
    hence contradiction by A1;
  end;
  thus v<>0.V
  proof
    assume v=0.V;
    then 1*v = 0.V by RLVECT_1:10;
    then 0.V + 1*v = 0.V by RLVECT_1:4;
    then 0*u + jj*v =0.V by RLVECT_1:10;
    hence contradiction by A1;
  end;
end;
