 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve s for FinSequence,
   V1,V2,V3 for finite-rank free Z_Module,
   f,f1,f2 for Function of V1,V2,
   g for Function of V2,V3,
   b1 for OrdBasis of V1,
   b2 for OrdBasis of V2,
   b3 for OrdBasis of V3,
   v1,v2 for Vector of V2,
   v,w for Element of V1;
 reserve p2,F for FinSequence of V1,
   p1,d for FinSequence of INT.Ring,
   KL for Linear_Combination of V1;

theorem VS10Th28:
  for V being non trivial free Z_Module,
  f being non constant 0-preserving Functional of V
  ex v being Vector of V st v <> 0.V & f.v <> 0.INT.Ring
  proof
    let V be non trivial free Z_Module,
    f be non constant 0-preserving Functional of V;
    A1: f.(0.V) = 0.INT.Ring by HAHNBAN1:def 9;
    assume
    A2: for v being Vector of V st v <> 0.V holds f.v = 0.INT.Ring;
    now
      let x, y be object;
      assume x in dom f & y in dom f;
      then reconsider v = x, w = y as Vector of V;
      thus f.x = f.v
      .= 0 by A2,A1
      .= f.w by A2,A1
      .= f.y;
    end;
    hence contradiction by FUNCT_1:def 10;
  end;
