
theorem Th129:
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2,
    W1 being F-defined Walk of G1, n being odd Element of NAT
  st n <= len W1 holds F_V.(W1.n) = (F.:W1).n
proof
  let G1, G2 be _Graph, F be non empty PGraphMapping of G1, G2;
  let W1 be F-defined Walk of G1, n be odd Element of NAT;
  assume A1: n <= len W1;
  then A2: n <= len (F.:W1) by Th125;
  A3: 1 <= (n+1) div 2 & (n+1) div 2 <= len W1.vertexSeq() by A1, GLIB_001:68;
  A4: (n+1) div 2 in dom W1.vertexSeq() by A3, FINSEQ_3:25;
  W1.n = W1.vertexAt(n) by A1, GLIB_001:def 8
    .= W1.vertexSeq().((n+1) div 2) by A1, GLIB_001:72;
  hence F_V.(W1.n) = (F_V * W1.vertexSeq()).((n+1) div 2) by A4, FUNCT_1:13
    .= ((F.:W1).vertexSeq()).((n+1) div 2) by Def37
    .= (F.:W1).vertexAt(n) by A2, GLIB_001:72
    .= (F.:W1).n by A2, GLIB_001:def 8;
end;
