
theorem Th7:
for m,n be non zero Nat, X be non-empty m-element FinSequence st n <= m holds
  (Pt2FinSeq X).n is Function of CarProduct SubFin(X,n),product SubFin(X,n)
proof
    let m,n be non zero Nat, X be non-empty m-element FinSequence;
    assume
A1: n <= m;
A2: 1 <= n by NAT_1:14;

    defpred P[Nat] means 1 <= $1 & $1 <= n implies
     ex i be non zero Nat st $1 = i &
      (Pt2FinSeq X).i is Function of
        CarProduct SubFin(X,i),product SubFin(X,i);

A3: P[0];
A4: for k be Nat st P[k] holds P[k+1]
    proof
     let k be Nat;
     assume
A5:  P[k];
     assume
A6:  1 <= k+1 & k+1 <= n;
     reconsider i = k+1 as non zero Nat;
     take i;
     thus k+1 = i;
     thus (Pt2FinSeq X).i is Function of
        CarProduct SubFin(X,i),product SubFin(X,i)
     proof
      per cases;
      suppose
A7:    k = 0;
       ex id1 be Function of CarProduct SubFin(X,1),product SubFin(X,1) st
       (Pt2FinSeq X).1 = id1 & id1 is bijective
     & for x be object st x in CarProduct SubFin(X,1) holds id1.x = <*x*>
         by Def5;
       hence (Pt2FinSeq X).i is Function of
        CarProduct SubFin(X,i),product SubFin(X,i) by A7;
      end;
      suppose k <> 0; then
       consider i0 be non zero Nat such that
A8:   k = i0
     & (Pt2FinSeq X).i0 is Function of
         CarProduct SubFin(X,i0),product SubFin(X,i0)
           by A5,A6,NAT_1:13,14;
       k < n by A6,NAT_1:13; then
A9:   i0 < m by A1,A8,XXREAL_0:2; then
       consider Fi be Function of CarProduct SubFin(X,i0),product SubFin(X,i0),
         IK be Function of [: CarProduct SubFin(X,i0),ElmFin(X,i0+1) :],
            product SubFin(X,i0+1) such that
A10:   Fi = (Pt2FinSeq X).i0 & IK = (Pt2FinSeq X).(i0+1) & Fi is bijective
     & IK is bijective
     & for x,y be object st x in CarProduct SubFin(X,i0) & y in ElmFin(X,i0+1)
        ex s be FinSequence st Fi.x = s & IK.(x,y) = s^<*y*> by Def5;

       CarProduct SubFin(X,i0+1) = [: CarProduct SubFin(X,i0),ElmFin(X,i0+1) :]
         by A9,MEASUR13:9;
       hence (Pt2FinSeq X).i is Function of
         CarProduct SubFin(X,i),product SubFin(X,i) by A8,A10;
      end;
     end;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2(A3,A4); then
    consider i be non zero Nat such that
A11: i = n
  & (Pt2FinSeq X).i is Function of
        CarProduct SubFin(X,i),product SubFin(X,i) by A2;
    thus thesis by A11;
end;
