
theorem
  for V, W being right_zeroed non empty ModuleStr over INT.Ring,
  f being FrForm of V,W st f is additiveFAF or f is additiveSAF
  holds f is constant iff
  for v being Vector of V, w being Vector of W holds f.(v,w) = 0.INT.Ring
  proof
    let V, W be right_zeroed non empty ModuleStr over INT.Ring,
    f be FrForm 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.INT.Ring by A4,HTh29;
      end;
      suppose
        A5: f is additiveSAF;
        thus f.(v,w) = f.(0.V,w) by A1,A3,BINOP_1:19
        .= 0.INT.Ring by A5,HTh30;
      end;
    end;
    hereby
      assume
      A6: for v being Vector of V, w being Vector of W holds
      f.(v,w) = 0.INT.Ring;
      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.INT.Ring by A6
        .= f.(s,t) by A6
        .= f.y by A10;
      end;
      hence f is constant by FUNCT_1:def 10;
    end;
  end;
