
theorem Th42: :: PMV(a,b,c) = PMV(b,a,c)
  for T being _Tree for a, b, c being Vertex of T holds
  MiddleVertex(a,b,c) = MiddleVertex(b,a,c)
proof
  let T be _Tree;
  let a, b, c be Vertex of T;
  set PMV1 = MiddleVertex(a,b,c);
  set PMV2 = MiddleVertex(b,a,c);
A1: T.pathBetween(a,b).vertices() = T.pathBetween(b,a).vertices() & T
  .pathBetween(b,c).vertices() = T.pathBetween(c,b).vertices() by Th32;
A2: T.pathBetween(c,a).vertices() = T.pathBetween(a,c).vertices() by Th32;
  T.pathBetween(a,b).vertices() /\ T.pathBetween(b,c).vertices() /\ T
  .pathBetween(c,a).vertices() = { PMV1 } & T.pathBetween(b,a).vertices() /\ T
  .pathBetween(a,c).vertices() /\ T.pathBetween(c,b).vertices() = { PMV2 } by
Def3;
  then {PMV1} = {PMV2} by A1,A2,XBOOLE_1:16;
  hence thesis by ZFMISC_1:3;
end;
