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 )

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

( 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 in dom f
; :: thesis: n + 1 <= len f
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;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

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