
theorem Th37:
  for T being _Tree for a, b, c being Vertex of T holds c in T
  .pathBetween(a,b).vertices() iff T.pathBetween(a,c) is_a_prefix_of T
  .pathBetween(a,b)
proof
  let T be _Tree, a, b, c be Vertex of T;
  hereby
    assume c in T.pathBetween(a,b).vertices();
    then
    T.pathBetween(a,b) = T.pathBetween(a,c).append(T.pathBetween(c,b)) by Th36;
    hence T.pathBetween(a,c) c= T.pathBetween(a,b) by Th15;
  end;
  assume T.pathBetween(a,c) c= T.pathBetween(a,b);
  then
A1: T.pathBetween(a,c).vertices() c= T.pathBetween(a,b).vertices() by Th13;
  c in T.pathBetween(a,c).vertices() by Th29;
  hence thesis by A1;
end;
