
theorem Th19:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  holds (the carrier of product X) --> 0.Y is MultilinearOperator of X,Y
  proof
    let X be RealNormSpace-Sequence,
        Y be RealNormSpace;
    set f0 = (the carrier of product X) --> 0.Y;
    now
      let i be Element of dom X, u be Element of product X;
      set F = f0 * reproj(i,u);
      A1: dom F = the carrier of X.i by FUNCT_2:def 1;
      A4: for z being object st z in dom F holds F.z = 0.Y
      proof
        let z being object;
        assume z in dom F; then
        A2: z in the carrier of X.i by FUNCT_2:def 1;
        thus F.z = f0.(reproj (i,u).z) by A2,FUNCT_2:15
        .= 0.Y by A2,FUNCT_2:5,FUNCOP_1:7;
      end;
      reconsider f = f0 * reproj(i,u) as Function of X.i,Y;
      A5: f is homogeneous
      proof
        let x be VECTOR of X.i, r be Real;
        thus f.(r*x) = r * 0.Y by A1,A4
        .= r * f.x by A1,A4;
      end;
      now
        let x,y be VECTOR of X.i;
        thus f.(x+y) = 0.Y by A1,A4
        .= f.x+0.Y by A1,A4
        .= f.x+f.y by A1,A4;
      end;
      hence f0 * reproj(i,u) is LinearOperator of X.i,Y
        by A5,VECTSP_1:def 20;
    end;
    hence thesis by Def3;
  end;
