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 (W1 is_Walk_from x,y iff W2 is_Walk_from x ,y)
proof
  assume
A1: W1 is Subwalk of W2;
  hereby
A2: W1 is_Walk_from W2.first(),W2.last() by A1,Def32;
    assume
A3: W1 is_Walk_from x,y;
    then W1.last() = y;
    then
A4: y = W2.last() by A2;
    W1.first() = x by A3;
    then x = W2.first() by A2;
    hence W2 is_Walk_from x,y by A4;
  end;
  assume
A5: W2 is_Walk_from x,y;
  then
A6: W2.last() = y;
  W2.first() = x by A5;
  hence thesis by A1,A6,Def32;
end;
