reserve X,Y,Z for non trivial RealBanachSpace;

theorem
  for X,Y,Z be RealNormSpace holds
  ex I be BilinearOperator of
            R_NormSpace_of_BoundedLinearOperators(X,Y),
            R_NormSpace_of_BoundedLinearOperators(Y,Z),
            R_NormSpace_of_BoundedLinearOperators(X,Z)
  st I is_continuous_on the carrier of
         [: R_NormSpace_of_BoundedLinearOperators(X,Y),
            R_NormSpace_of_BoundedLinearOperators(Y,Z):]
   & for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
         v be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z)
     holds I.(u,v) = v*u
  proof
    let X,Y,Z be RealNormSpace;
    set E = R_NormSpace_of_BoundedLinearOperators(X,Y);
    set F = R_NormSpace_of_BoundedLinearOperators(Y,Z);
    set G = R_NormSpace_of_BoundedLinearOperators(X,Z);
    defpred P1[object,object] means
    ex u be Point of E,v be Point of F
    st $1=[u,v] & $2 = v*u;
    A1: for x being object st x in the carrier of [:E,F:] holds
        ex y being object st y in the carrier of G & P1[x,y]
    proof
      let x be object;
      assume x in the carrier of [:E,F:]; then
      consider u be Point of E,v be Point of F such that
      A2: x = [u,v] by PRVECT_3:18;
      take y = v*u;
      thus y in the carrier of G & P1[x,y] by A2;
    end;
    consider L being Function of the carrier of [:E,F:],the carrier of G
    such that
    A3: for x being object st x in the carrier of [:E,F:]
        holds P1[x,L . x] from FUNCT_2:sch 1(A1);
    A4: for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y),
            v be Point of R_NormSpace_of_BoundedLinearOperators(Y,Z)
        holds L.(u,v) = v*u
    proof
      let u be Point of E, v be Point of F;
      consider u1 be Point of E, v1 be Point of F such that
      A5: [u,v] = [u1,v1] & L.[u,v] = v1*u1 by A3;
      u = u1 & v = v1 by A5,XTUPLE_0:1;
      hence thesis by A5;
    end;
    A6: for x1,x2 be Point of E, y be Point of F
        holds L.(x1+x2,y) = L.(x1,y) + L.(x2,y)
    proof
      let x1,x2 be Point of E, y be Point of F;
      thus L.(x1+x2,y) = y*(x1+x2) by A4
      .= y*x1 + y*x2 by LM100
      .= L.(x1,y) + y*x2 by A4
      .= L.(x1,y) + L.(x2,y) by A4;
    end;
    A7: for x be Point of E, y be Point of F, a be Real
        holds L.(a*x,y) = a*L.(x,y)
    proof
      let x be Point of E, y be Point of F, a be Real;
      thus L.(a*x,y) = y*(a*x) by A4
      .= a*y*x by LOPBAN1624
      .= a*(y*x) by LOPBAN1624
      .= a*L.(x,y) by A4;
    end;
    A8: for x be Point of E, y1,y2 be Point of F
        holds L.(x,y1+y2) = L.(x,y1) + L.(x,y2)
    proof
      let x be Point of E, y1,y2 be Point of F;
      thus L. (x,y1+y2) = (y1+y2) * x by A4
      .= y1*x + y2*x by LM200
      .= L.(x,y1) + y2*x by A4
      .= L.(x,y1) + L.(x,y2) by A4;
    end;
    for x be Point of E, y be Point of F, a be Real
    holds L.(x,a*y) = a*L.(x,y)
    proof
      let x be Point of E, y be Point of F, a be Real;
      thus L. (x,a*y) = (a*y)*x by A4
      .= a*(y*x) by LOPBAN1624
      .= a*L.(x,y) by A4;
    end; then
    reconsider L as BilinearOperator of E,F,G by A6,A7,A8,LOPBAN_8:12;
    take L;
    set K = 1;
    for x be Point of E,y be Point of F
    holds ||.L. (x,y) .|| <= K * ||.x.|| * ||.y.||
    proof
      let x be Point of E,y be Point of F;
      L.(x,y) = y*x by A4;
      hence ||.L. (x,y) .|| <= K * ||.x.|| * ||.y.|| by NRM;
    end;
    hence thesis by A4,LOPBAN_8:13;
  end;
