reserve G,G1,G2 for _Graph;
reserve W,W1,W2 for Walk of G;
reserve e,x,y,z for set;
reserve v for Vertex of G;
reserve n,m for Element of NAT;

theorem
  W1 is Subwalk of W2 implies for m being even Element of NAT st 1 <= m
& m <= len W1 holds ex n being even Element of NAT st m <= n & n <= len W2 & W1
  .m = W2.n
proof
  assume W1 is Subwalk of W2;
  then
A1: ex es being Subset of W2.edgeSeq() st W1.edgeSeq() = Seq es by Def32;
  let m be even Element of NAT such that
A2: 1 <= m and
A3: m <= len W1;
A4: W1.m = W1.edgeSeq().(m div 2) by A2,A3,Lm40;
  m div 2 in dom W1.edgeSeq() by A2,A3,Lm40;
  then consider ndiv2 being Element of NAT such that
A5: ndiv2 in dom W2.edgeSeq() and
A6: m div 2 <= ndiv2 and
A7: W1.m = W2.edgeSeq().ndiv2 by A1,A4,Th3;
A8: ndiv2 <= len W2.edgeSeq() by A5,FINSEQ_3:25;
  2 divides m by PEPIN:22;
  then
A9: 2 * (m div 2) = m by NAT_D:3;
  2*ndiv2 in dom W2 by A5,Lm41;
  then
A10: 2*ndiv2 <= len W2 by FINSEQ_3:25;
  1 <= ndiv2 by A5,FINSEQ_3:25;
  then W1.m = W2.(2*ndiv2) by A7,A8,Def15;
  hence thesis by A6,A9,A10,XREAL_1:64;
end;
