
theorem Th51:
  for T being _Tree for P1, P2 be Path of T for a, b, c being
Vertex of T st P1 = T.pathBetween(a,b) & P2 = T.pathBetween(a,c) & not b in P2
  .vertices() & not c in P1.vertices() holds MiddleVertex(a, b, c) = P1.len
  maxPrefix(P1,P2)
proof
  let T be _Tree, P1, P2 be Path of T, a, b, c be Vertex of T such that
A1: P1 = T.pathBetween(a,b) & P2 = T.pathBetween(a,c) and
A2: ( not b in P2.vertices())& not c in P1.vertices();
  ( not P1 c= P2)& not P2 c= P1 by A1,A2,Th37;
  then
A3: T.pathBetween(a,b).vertices() /\ T.pathBetween(b,c).vertices() /\ T
  .pathBetween(c,a).vertices() = { P1.len maxPrefix(P1,P2) } by A1,Lm2;
  T.pathBetween(a,b).vertices() /\ T.pathBetween(b,c).vertices() /\ T
  .pathBetween(c,a).vertices() = { MiddleVertex(a, b, c) } by Def3;
  hence thesis by A3,ZFMISC_1:3;
end;
