
theorem Th9:
for m,n be non zero Nat, X be non-empty m-element FinSequence
 st n < m holds
 CarProduct SubFin(X,n+1) = [: CarProduct SubFin(X,n),ElmFin(X,n+1) :]
proof
    let m,n be non zero Nat, X be non-empty m-element FinSequence;
    assume n < m; then
A1: n <= n+1 & n+1 <= m by NAT_1:13; then
A2: ElmFin(SubFin(X,n+1),n+1) = ElmFin(X,n+1) by Th8;

    CarProduct SubFin(X,n+1)
     = [: CarProduct SubFin(SubFin(X,n+1),n),ElmFin(SubFin(X,n+1),n+1) :]
       by Th6;
    hence CarProduct SubFin(X,n+1)
     = [: CarProduct SubFin(X,n),ElmFin(X,n+1) :] by A1,A2,Th7;
end;
