
theorem Th5:
for m,n be non zero Nat, X be non-empty m-element FinSequence st n < m holds
  (ProdFinSeq X).(n+1) = [: (ProdFinSeq SubFin(X,n)).n,ElmFin(X,n+1) :]
proof
    let m,n be non zero Nat, X be non-empty m-element FinSequence;
    assume
A1:  n < m; then
A2: n+1 <= m by NAT_1:13;
    (ProdFinSeq X).(n+1) = [: (ProdFinSeq X).n, X.(n+1) :] by A1,Def3
     .= [: (ProdFinSeq SubFin(X,n)).n, X.(n+1) :] by A1,Th4;
    hence (ProdFinSeq X).(n+1)
     = [: (ProdFinSeq SubFin(X,n)).n,ElmFin(X,n+1) :] by A2,Def1;
end;
