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

theorem IS02:
  for u be BilinearOperator of E,F,G
  holds u * (IsoCPRLSP (E,F))" is MultilinearOperator of <*E,F*>,G
  proof
    let u be BilinearOperator of E,F,G;
    reconsider M = u * (IsoCPRLSP (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_3:14;
      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,NDIFF5def4
          .= <*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_3:14;
          A18: L.x1 = M.((reproj(i,s)).x1) by A8,FUNCT_1:13
          .= (u * (IsoCPRLSP (E,F))").(<*x1,y*>) by A9
          .= u.((IsoCPRLSP (E,F))" .x1y) by FUNCT_2:15
          .= u.(x1,y) by defISOA1;
          A19: L.x2 = M.((reproj(i,s)).x2) by A8,FUNCT_1:13
          .= (u * (IsoCPRLSP(E,F))").(<*x2,y*>) by A9
          .= u.((IsoCPRLSP (E,F))" .x2y) by FUNCT_2:15
          .= u.(x2,y) by defISOA1;
          L.(x1+x2) = M.((reproj (i,s)).(x1+x2)) by A8,FUNCT_1:13
          .= (u * (IsoCPRLSP(E,F))") .(<*(x1+x2),y*>) by A9
          .= u.((IsoCPRLSP(E,F))" .(x12y)) by FUNCT_2:15
          .= u.(x1+x2,y) by defISOA1;
          hence L.(x1+x2) = L.x1 + L.x2 by A18,A19,LOPBAN_8:11;
        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_3:14;
          A23: L.x1 = M.((reproj(i,s)).x1) by A8,FUNCT_1:13
          .= (u * (IsoCPRLSP(E,F))").(<*x1,y*>) by A9
          .= u.((IsoCPRLSP(E,F))" .x1y) by FUNCT_2:15
          .= u.(x1,y) by defISOA1;
          L.(a*x1) = M.((reproj (i,s)).(a*x1)) by A8,FUNCT_1:13
          .= (u *(IsoCPRLSP(E,F))").(<*a*x1,y*>) by A9
          .= u.((IsoCPRLSP(E,F))" .ax1y) by FUNCT_2:15
          .= u.(a*x1,y) by defISOA1;
          hence L.(a*x1) = a * L.x1 by A23,LOPBAN_8:11;
        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,NDIFF5def4
          .= <*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_3:14;
          A37: L.y1 = M.((reproj(i,s)).y1) by A27,FUNCT_1:13
          .= (u *(IsoCPRLSP(E,F))").(<*x,y1*>) by A28
          .= u.((IsoCPRLSP(E,F))" .(y1y)) by FUNCT_2:15
          .= u.(x,y1) by defISOA1;
          A38: L.y2 = M.((reproj(i,s)).y2) by A27,FUNCT_1:13
          .= (u * (IsoCPRLSP(E,F))").(<*x,y2*>) by A28
          .= u.((IsoCPRLSP(E,F))" .(y2y)) by FUNCT_2:15
          .= u.(x,y2) by defISOA1;
          L.(y1+y2) = M.((reproj(i,s)).(y1+y2)) by A27,FUNCT_1:13
          .= (u * (IsoCPRLSP(E,F))").(<*x,(y1+y2)*>) by A28
          .= u.((IsoCPRLSP(E,F))" .(y12y)) by FUNCT_2:15
          .= u.(x,y1+y2) by defISOA1;
          hence L.(y1+y2) = L.y1 + L.y2 by A37,A38,LOPBAN_8:11;
        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_3:14;
          A42: L.y1 = M.((reproj (i,s)).y1) by A27,FUNCT_1:13
          .= (u * (IsoCPRLSP(E,F))").(<*x,y1*>) by A28
          .= u.((IsoCPRLSP(E,F))" .x1y) by FUNCT_2:15
          .= u.(x,y1) by defISOA1;
          L.(a*y1) = M.((reproj(i,s)).(a*y1)) by A27,FUNCT_1:13
          .= (u * (IsoCPRLSP(E,F))").(<*x,a*y1*>) by A28
          .= u.((IsoCPRLSP (E,F))" .(ax1y)) by FUNCT_2:15
          .= u.(x, a*y1) by defISOA1;
          hence L.(a*y1) = a * L.y1 by A42,LOPBAN_8:11;
        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;
    hence thesis by LOPBAN10:def 3;
  end;
