let G be _Graph; :: thesis: for W being Walk of G
for n being Nat holds
( n in dom (W .edgeSeq()) iff 2 * n in dom W )
let W be Walk of G; :: thesis: for n being Nat holds
( n in dom (W .edgeSeq()) iff 2 * n in dom W )
let n be Nat; :: thesis: ( n in dom (W .edgeSeq()) iff 2 * n in dom W )
hereby :: thesis: ( 2 * n in dom W implies n in dom (W .edgeSeq()) )
assume A1: n in dom (W .edgeSeq()) ; :: thesis: 2 * n in dom W
then n <= len (W .edgeSeq()) by FINSEQ_3:25;
then 2 * n <= (len (W .edgeSeq())) * 2 by NAT_1:4;
then 2 * n <= ((len (W .edgeSeq())) * 2) + 1 by NAT_1:12;
then A2: 2 * n <= len W by Def15;
1 <= n by A1, FINSEQ_3:25;
then 1 <= n + n by NAT_1:12;
hence 2 * n in dom W by A2, FINSEQ_3:25; :: thesis: verum
end;
assume A3: 2 * n in dom W ; :: thesis: n in dom (W .edgeSeq())
then A4: 2 * n <= len W by FINSEQ_3:25;
1 <= 2 * n by A3, FINSEQ_3:25;
then (2 * n) div 2 in dom (W .edgeSeq()) by A4, Lm40;
hence n in dom (W .edgeSeq()) by NAT_D:18; :: thesis: verum