
theorem
  for G1, G2 being _Graph, F being non empty PGraphMapping of G1, G2,
    W1 being F-defined Walk of G1, n being even Element of NAT
  st 1 <= n & n <= len W1 holds F_E.(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 even Element of NAT;
  assume A1: 1 <= n & n <= len W1;
  then A2: n <= len (F.:W1) by Th125;
  A3: n div 2 in dom W1.edgeSeq() & W1.n = W1.edgeSeq().(n div 2)
    by A1, GLIB_001:77;
  thus F_E.(W1.n) = (F_E * W1.edgeSeq()).(n div 2) by A3, FUNCT_1:13
    .= ((F.:W1).edgeSeq()).(n div 2) by Def37
    .= (F.:W1).n by A1, A2, GLIB_001:77;
end;
