 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
  for V, W being right_zeroed non empty ModuleStr over INT.Ring,
      f being Form of V,W
  st f is additiveFAF or f is additiveSAF
  holds f is constant iff
  for v be Vector of V, w be Vector of W holds f.(v,w)=0
  proof
    let V, W be right_zeroed non empty ModuleStr over INT.Ring,
      f be Form of V,W;
    A1: dom f = [: the carrier of V, the carrier of W:] by FUNCT_2:def 1;
    assume
    A2: f is additiveFAF or f is additiveSAF;
    hereby
      assume
      A3: f is constant;
      let v be Vector of V, w be Vector of W;
      per cases by A2;
      suppose
        A4: f is additiveFAF;
        thus f.(v,w) = f.(v,0.W) by A1,A3,BINOP_1:19
        .= 0 by A4,BLTh29;
      end;
      suppose
        A5: f is additiveSAF;
        thus f.(v,w) = f.(0.V,w) by A1,A3,BINOP_1:19
        .= 0 by A5,BLTh30;
      end;
    end;
    hereby
      assume
      A6: for v being Vector of V, w being Vector of W holds f.(v,w) = 0;
      now
        let x, y be object such that
        A7: x in dom f and
        A8: y in dom f;
        consider v be Vector of V, w be Vector of W such that
        A9: x = [v,w] by A7,DOMAIN_1:1;
        consider s be Vector of V, t be Vector of W such that
        A10: y = [s,t] by A8,DOMAIN_1:1;
        thus f.x = f.(v,w) by A9
        .= 0 by A6
        .=f.(s,t) by A6
        .=f.y by A10;
      end;
      hence f is constant;
    end;
  end;
