
theorem Th40:
  for K be add-associative right_zeroed right_complementable non
  empty doubleLoopStr, V,W be right_zeroed non empty ModuleStr over K, f be
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.K
proof
  let K be add-associative right_zeroed right_complementable non empty
doubleLoopStr, V,W be right_zeroed non empty ModuleStr over K, 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.K by A4,Th29;
    end;
    suppose
A5:   f is additiveSAF;
      thus f.(v,w) = f.(0.V,w) by A1,A3,BINOP_1:19
        .= 0.K by A5,Th30;
    end;
  end;
  hereby
    assume
A6: for v be Vector of V, w be Vector of W holds f.(v,w)=0.K;
    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.K by A6
        .=f.(s,t) by A6
        .=f.y by A10;
    end;
    hence f is constant by FUNCT_1:def 10;
  end;
end;
