let n be Nat; :: thesis: for f being FinSequence of REAL holds
( n + 1 <= len f iff n + 1 in dom f )
let f be FinSequence of REAL ; :: thesis: ( n + 1 <= len f iff n + 1 in dom f )
thus ( n + 1 <= len f implies n + 1 in dom f ) :: thesis: ( n + 1 in dom f implies n + 1 <= len f )

proof

assume n + 1 <= len f ; :: thesis: n + 1 in dom f
then n < len f by NAT_1:13;
then n + 1 in Seg (len f) by Th2;
hence n + 1 in dom f by FINSEQ_1:def 3; :: thesis: verum

end;

assume n + 1 in dom f ; :: thesis: n + 1 <= len f
then n + 1 in Seg (len f) by FINSEQ_1:def 3;
then n < len f by Th2;
hence n + 1 <= len f by NAT_1:13; :: thesis: verum