
theorem Th41:
  for n be Nat,
      A,B be RealNormSpace-Sequence,
      X,Y be RealNormSpace
  st len A = n + 1 & A|n = B & X = A.(n + 1)
  holds NestingLB(A,Y)
      = R_NormSpace_of_BoundedLinearOperators(X,NestingLB(B,Y))
  proof
    let n be Nat,
        A,B be RealNormSpace-Sequence,
        X,Y be RealNormSpace;
    assume
    A1: len A = n + 1 & A|n = B & X = A.(n + 1);

    consider f be Function such that
    A2: dom f = NAT
      & NestingLB(A,Y) = f.(len A)
      & f.0 = Y
      & for j be Nat st j < len A holds
        ex V be RealNormSpace, k be Element of dom A
        st V = f.j & j + 1 = k
         & f.(j + 1) = R_NormSpace_of_BoundedLinearOperators(A.k,V) by Def3;

    A3: len B = n by A1,FINSEQ_3:53;
    then len B < len A by A1,NAT_1:13;
    then consider V be RealNormSpace, k be Element of dom A
    such that
    A4: V = f.(len B) & (len B) + 1 = k
      & f.((len B) + 1) = R_NormSpace_of_BoundedLinearOperators(A.k,V) by A2;

    A5: for j be Nat st j < len B holds
        ex V be RealNormSpace, k be Element of dom B
        st V = f.j & j + 1 = k
         & f.(j + 1) = R_NormSpace_of_BoundedLinearOperators(B.k,V)
    proof
      let j be Nat;
      assume
      A6: j < len B;
      then j < len A by A1,A3,NAT_1:13;
      then consider V be RealNormSpace, k be Element of dom A
      such that
      A7: V = f.j & j + 1 = k
        & f.(j + 1) = R_NormSpace_of_BoundedLinearOperators(A.k,V) by A2;

      A8: j + 1 <= len B by A6,NAT_1:13;
      1 <= j + 1 by NAT_1:11;
      then
      A9: j + 1 in Seg len B by A8;
      then reconsider k1 = j + 1 as Element of dom B by FINSEQ_1:def 3;
      A.k = B.k1 by A1,A7,A9,FUNCT_1:49,A3;
      hence thesis by A7;
    end;

    A10: A.k = X by A1,A4,FINSEQ_3:53;

    thus NestingLB(A,Y)
     = f.k by A1,A2,A4,FINSEQ_3:53
    .= R_NormSpace_of_BoundedLinearOperators(X,NestingLB(B,Y))
      by A2,A4,A5,A10,Def3;
  end;
