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 Th22:
 for x,y being object holds
  W is_Walk_from x,y iff W.reverse() is_Walk_from y, x
proof let x,y be object;
A1: len W = len W.reverse() by FINSEQ_5:def 3;
  thus W is_Walk_from x, y implies W.reverse() is_Walk_from y,x
  by A1,FINSEQ_5:62;
  assume
A2: W.reverse() is_Walk_from y,x;
  then W.reverse().1=y;
  then
A3: W.(len W) = y by FINSEQ_5:62;
  W.reverse().(len W.reverse())=x by A2;
  then W.1 = x by A1,FINSEQ_5:62;
  hence thesis by A3;
end;
