 reserve X,Y,Z,E,F,G,S,T for RealLinearSpace;
 reserve X,Y,Z,E,F,G for RealNormSpace;
 reserve S,T for RealNormSpace-Sequence;

theorem IS02A:
  for u be Lipschitzian BilinearOperator of E,F,G
  holds u * (IsoCPNrSP(E,F))" is Lipschitzian MultilinearOperator of <*E,F*>,G
  proof
    let u be Lipschitzian BilinearOperator of E,F,G;
    reconsider M = u * (IsoCPNrSP(E,F))"
      as Function of product <*E,F*>,G;
    A1: dom <*E,F*> = Seg len <*E,F*> by FINSEQ_1:def 3
    .= {1,2} by FINSEQ_1:2,44;
    then reconsider i1=1, i2=2 as Element of dom <*E,F*> by TARSKI:def 2;
    for i be Element of dom <*E,F*>,
        s be Element of product <*E,F*> holds
    M * reproj(i,s) is LinearOperator of <*E,F*>.i,G
    proof
      let i be Element of dom <*E,F*>,
          s be Element of product <*E,F*>;
      consider x be Point of E, y be Point of F such that
      A3: s = <*x,y*> by PRVECT_3:19;
      len s = 2 by A3,FINSEQ_1:44; then
      A5: dom s = {1,2} by FINSEQ_1:2,def 3;
      per cases by A1,TARSKI:def 2;
      suppose
        A6: i = 1; then
        A7: <*E,F*>.i = E; then
        reconsider L = M * reproj(i,s) as Function of E,G;
        A8: dom reproj(i,s) = the carrier of <*E,F*>.i by FUNCT_2:def 1
        .= the carrier of E by A6;
        A9: for z be Point of E holds reproj(i,s).z = <*z,y*>
        proof
          let x1 be Point of E;
          A10: len (s +* (i,x1)) = 2 by LemmaA;
          A11: (s +* (i,x1)).1 = x1 by A1,A5,A6,FUNCT_7:31;
          A12: (s +* (i,x1)).2 = s.2 by A6,FUNCT_7:32
          .= y by A3;
          thus reproj(i,s).x1 = s +* (i,x1) by A7,NDIFF_5:def 4
          .= <*x1,y*> by A10,A11,A12,FINSEQ_1:44;
        end;
        A14: for x1,x2 be Point of E holds L.(x1+x2) = L.x1 + L.x2
        proof
          let x1,x2 be Point of E;
          reconsider x1y = <*x1,y*>, x2y = <*x2,y*>, x12y = <*x1+x2,y*>
            as Point of product <*E,F*> by PRVECT_3:19;
          A18: L.x1 = M.((reproj (i,s)).x1) by A8,FUNCT_1:13
          .= (u * (IsoCPNrSP(E,F))").(<*x1,y*>) by A9
          .= u.((IsoCPNrSP(E,F))" .(x1y)) by FUNCT_2:15
          .= u.(x1,y) by NDIFF_7:18;
          A19: L.x2 = M.((reproj(i,s)).x2) by A8,FUNCT_1:13
          .= (u * (IsoCPNrSP(E,F))").(<*x2,y*>) by A9
          .= u.((IsoCPNrSP(E,F))" .(x2y)) by FUNCT_2:15
          .= u.(x2,y) by NDIFF_7:18;
          L.(x1+x2) = M.((reproj(i,s)).(x1+x2)) by A8,FUNCT_1:13
          .= (u * (IsoCPNrSP(E,F))").(<*(x1+x2),y*>) by A9
          .= u.((IsoCPNrSP(E,F))" .(x12y)) by FUNCT_2:15
          .= u.(x1+x2,y) by NDIFF_7:18;
          hence L.(x1+x2) = L.x1 + L.x2 by A18,A19,LOPBAN_8:12;
        end;
        for x1 be Point of E, a be Real holds L.(a*x1) = a * L.x1
        proof
          let x1 be Point of E, a be Real;
          reconsider ax1y = <*a*x1,y*>, x1y = <*x1,y*>
            as Point of product <*E,F*> by PRVECT_3:19;
          A23: L.x1 = M.((reproj(i,s)).x1) by A8,FUNCT_1:13
          .= (u * (IsoCPNrSP(E,F))").(<*x1,y*>) by A9
          .= u.((IsoCPNrSP(E,F))" .x1y) by FUNCT_2:15
          .= u.(x1,y) by NDIFF_7:18;
          L.(a*x1) = M.((reproj(i,s)).(a*x1)) by A8,FUNCT_1:13
          .= (u * (IsoCPNrSP(E,F))").(<*a*x1,y*>) by A9
          .= u.((IsoCPNrSP(E,F))" .(ax1y)) by FUNCT_2:15
          .= u.(a*x1,y) by NDIFF_7:18;
          hence L.(a*x1) = a * L.x1 by A23,LOPBAN_8:12;
        end;
        hence M * reproj(i,s) is LinearOperator of <*E,F*>.i,G
          by A7,A14,LOPBAN_1:def 5,VECTSP_1:def 20;
      end;
      suppose
        A25: i = 2; then
        A26: <*E,F*>.i = F; then
        reconsider L = M * reproj(i,s) as Function of F,G;
        A27: dom reproj(i,s) = the carrier of <*E,F*> .i by FUNCT_2:def 1
         .= the carrier of F by A25;
        A28: for z be Point of F holds reproj(i,s).z = <*x,z*>
        proof
          let y1 be Point of F;
          A29: len (s +* (i,y1)) = 2 by LemmaA;
          A30: (s +* (i,y1)).2 = y1 by A1,A5,A25,FUNCT_7:31;
          A31: (s +* (i,y1)).1 = s.1 by A25,FUNCT_7:32
          .= x by A3;
          thus reproj(i,s).y1 = s +* (i,y1) by A26,NDIFF_5:def 4
          .= <*x,y1*> by A29,A30,A31,FINSEQ_1:44;
        end;
        A33: for y1,y2 be Point of F holds L.(y1+y2) = L.y1 + L.y2
        proof
          let y1,y2 be Point of F;
          reconsider y1y = <*x,y1*>, y2y = <*x,y2*>, y12y = <*x,y1+y2*>
            as Point of product <*E,F*> by PRVECT_3:19;
          A37: L.y1 = M.((reproj (i,s)).y1) by A27,FUNCT_1:13
          .= (u * (IsoCPNrSP (E,F))").(<*x,y1*>) by A28
          .= u.((IsoCPNrSP(E,F))" .(y1y)) by FUNCT_2:15
          .= u.(x,y1) by NDIFF_7:18;
          A38: L.y2 = M.((reproj(i,s)).y2) by A27,FUNCT_1:13
          .= (u * (IsoCPNrSP(E,F))").(<*x,y2*>) by A28
          .= u.((IsoCPNrSP(E,F))" .(y2y)) by FUNCT_2:15
          .= u.(x,y2) by NDIFF_7:18;
          L.(y1+y2) = M.((reproj(i,s)).(y1+y2)) by A27,FUNCT_1:13
          .= (u * (IsoCPNrSP (E,F))").(<*x,(y1+y2)*>) by A28
          .= u.((IsoCPNrSP(E,F))" .(y12y)) by FUNCT_2:15
          .= u.(x,y1+y2) by NDIFF_7:18;
          hence L.(y1+y2) = L.y1 + L.y2 by A37,A38,LOPBAN_8:12;
        end;
        for y1 be Point of F, a be Real holds L.(a*y1) = a * L.y1
        proof
          let y1 be Point of F, a be Real;
          reconsider ax1y = <*x,a*y1*>, x1y = <*x,y1*>
            as Point of product <*E,F*> by PRVECT_3:19;
          A42: L.y1 = M.((reproj(i,s)).y1) by A27,FUNCT_1:13
          .= (u * (IsoCPNrSP(E,F))").(<*x,y1*>) by A28
          .= u.((IsoCPNrSP (E,F))" .x1y) by FUNCT_2:15
          .= u.(x,y1) by NDIFF_7:18;
          L.(a*y1) = M.((reproj (i,s)).(a*y1)) by A27,FUNCT_1:13
          .= (u * (IsoCPNrSP (E,F))").(<*x,a*y1*>) by A28
          .= u.((IsoCPNrSP(E,F))" .(ax1y)) by FUNCT_2:15
          .= u.(x,a*y1) by NDIFF_7:18;
          hence L.(a*y1) = a * L.y1 by A42,LOPBAN_8:12;
        end;
        hence M * reproj(i,s) is LinearOperator of <*E,F*>.i,G
          by A26,A33,LOPBAN_1:def 5,VECTSP_1:def 20;
      end;
    end; then
    reconsider M as MultilinearOperator of <*E,F*>,G by LOPBAN10:def 6;
    ex K being Real st 0 <= K &
    for s being Point of product <*E,F*> holds ||. M.s .|| <= K * NrProduct s
    proof
      consider K being Real such that
      A44: 0 <= K
       & for x be Point of E, y be Point of F
         holds ||. u.(x,y) .|| <= K * ||.x.|| * ||.y.|| by LOPBAN_9:def 3;
      take K;
      thus 0 <= K by A44;
      let xy be Point of product <*E,F*>;
      consider x be Point of E, y be Point of F such that
      A45: xy = <*x,y*> by PRVECT_3:19;
      A46: M.xy = u.((IsoCPNrSP(E,F)").xy) by FUNCT_2:15
      .= u.(x,y) by A45,NDIFF_7:18;
      consider Nx be FinSequence of REAL such that
      A47: dom Nx = dom <*E,F*>
      & (for i be Element of dom <*E,F*> holds Nx.i = ||.xy.i.||)
      & NrProduct xy = Product Nx by LOPBAN10:def 9;
      dom Nx = Seg len <*E,F*> by A47,FINSEQ_1:def 3
      .= Seg 2 by FINSEQ_1:44; then
      A48: len Nx = 2 by FINSEQ_1:def 3;
      A50: Nx.1 = ||.xy.i1.|| by A47
      .= ||.x.|| by A45;
      Nx.2 = ||.xy.i2.|| by A47
      .= ||.y.|| by A45; then
      Nx = <* ||.x.||,||.y.|| *> by A48,A50,FINSEQ_1:44; then
      A51: NrProduct xy = ||.x.|| * ||.y.|| by A47,RVSUM_1:99;
      ||. u.(x,y) .|| <= K * ||.x.|| * ||.y.|| by A44;
      hence thesis by A46,A51;
    end;
    hence thesis by LOPBAN10:def 10;
  end;
