
theorem ThTrivialLat1:
  for V being finite-rank free Z_Module, f being Function of
  [:the carrier of (0).V, the carrier of (0).V:], the carrier of F_Real
  st f = [:the carrier of (0).V, the carrier of (0).V:] --> 0.F_Real holds
  GenLat ((0).V,f) is Z_Lattice-like
  proof
    let V be finite-rank free Z_Module,
    f be Function of [:the carrier of (0).V, the carrier of (0).V:],
    the carrier of F_Real such that
    P1: f = [:the carrier of (0).V, the carrier of (0).V:] --> 0.F_Real;
    set X = GenLat ((0).V,f);
    reconsider X as Abelian add-associative right_zeroed
    right_complementable scalar-distributive vector-distributive
    scalar-associative scalar-unital
    non empty LatticeStr over INT.Ring;
    reconsider X as free Abelian add-associative right_zeroed
    right_complementable scalar-distributive vector-distributive
    scalar-associative scalar-unital
    non empty LatticeStr over INT.Ring;
    reconsider X as finite-rank free Abelian add-associative right_zeroed
    right_complementable scalar-distributive vector-distributive
    scalar-associative scalar-unital non empty LatticeStr over INT.Ring;
    P2: for x being Vector of X
    st for y being Vector of X holds <; x, y ;> = 0
    holds x = 0.X
    proof
      let x be Vector of X such that
      for y being Vector of X holds <; x, y ;> = 0;
      x in the carrier of (0).V;
      then x in {0.V} by VECTSP_4:def 3;
      hence x = 0.V by TARSKI:def 1
      .= 0.X by ZMODUL01:26;
    end;
    for x, y, z being Vector of X, a being Element of INT.Ring holds
    <; x, y ;> = <; y, x ;>
    & <; x+y, z ;> = <; x, z ;> + <; y, z ;>
    & <; a*x, y ;> = a * <; x, y ;>
     proof
      let x, y, z be Vector of X;
      let a be Element of INT.Ring;
      A1: the carrier of (0).V = {0.V} by VECTSP_4:def 3;
      then A3: 0.V in the carrier of (0).V by TARSKI:def 1;
      thus <; x, y ;> = <; y, x ;>
      proof
        x = 0.V & y = 0.V by A1,TARSKI:def 1;
        hence thesis;
      end;
      A4: f.[0.V, 0.V] = 0 by A3,P1,FUNCOP_1:7,ZFMISC_1:87;
      thus <; x+y, z ;> = <; x, z ;> + <; y, z ;>
      proof
        reconsider u = x, v = y, w = z as Vector of (0).V;
        B2: w = 0.V by A1,TARSKI:def 1;
        u = 0.V & v = 0.V by A1,TARSKI:def 1;
        hence thesis by A1,A4,B2,TARSKI:def 1;
      end;
      thus <; a*x, y ;> = a * <; x, y ;>
      proof
        reconsider u = x, v = y as Vector of (0).V;
        u = 0.V & v = 0.V by A1,TARSKI:def 1;
        hence thesis by A1,A4,TARSKI:def 1;
      end;
    end;
    hence thesis by P2;
  end;
