
theorem EQSET:
  for X, Y, Z be RealNormSpace, z be object holds
    ( z in BilinearOperators(X,Y,Z)
  iff
    z is BilinearOperator of X,Y,Z )
  proof
    let X, Y, Z be RealNormSpace, z be object;
    reconsider X0 = X,Y0=Y,Z0=Z as RealLinearSpace;
    hereby
      assume z in BilinearOperators(X,Y,Z); then
      z is BilinearOperator of X0,Y0,Z0 by Def6; then
      consider g be Function of
      [:the carrier of X0,the carrier of Y0:], the carrier of Z0
      such that
      A1: z = g & g is Bilinear by LOPBAN_8:def 2;
      thus z is BilinearOperator of X,Y,Z by A1,LOPBAN_8:def 3;
    end;
    assume z is BilinearOperator of X,Y,Z; then
    consider g be Function of
    [:the carrier of X,the carrier of Y:], the carrier of Z such that
    A1: z=g & g is Bilinear by LOPBAN_8:def 3;
    reconsider w=g as BilinearOperator of X0,Y0,Z0
      by LOPBAN_8:def 2,A1;
    w in BilinearOperators(X,Y,Z) by Def6;
    hence z in BilinearOperators(X,Y,Z) by A1;
  end;
