
theorem :: PMV302:
  for T being _Tree for P1, P2, P3, P4 be Path of T for a, b, c being
  Vertex of T st P1 = T.pathBetween(a,b) & P2 = T.pathBetween(a,c) & P3 = T
.pathBetween(b,a) & P4 = T.pathBetween(b,c) & not b in P2.vertices() & not c in
  P1.vertices() & not a in P4.vertices() holds P1.len maxPrefix(P1,P2) = P3.len
  maxPrefix(P3,P4)
proof
  let T be _Tree, P1, P2, P3, P4 be Path of T, a, b, c be Vertex of T such
  that
A1: P1 = T.pathBetween(a,b) and
A2: P2 = T.pathBetween(a,c) and
A3: P3 = T.pathBetween(b,a) and
A4: P4 = T.pathBetween(b,c) and
A5: not b in P2.vertices() and
A6: not c in P1.vertices() and
A7: not a in P4.vertices();
  now
    assume P4 c= P3;
    then
A8: P4.vertices() c= P3.vertices() by Th13;
    c in P4.vertices() by A4,Th29;
    then c in P3.vertices() by A8;
    hence contradiction by A1,A3,A6,Th32;
  end;
  then not c in P3.vertices() by A3,A4,Th37;
  then
A9: MiddleVertex(b,a,c) = P3.len maxPrefix(P3,P4) by A3,A4,A7,Th51;
  MiddleVertex(a,b,c) = P1.len maxPrefix(P1,P2) by A1,A2,A5,A6,Th51;
  hence thesis by A9,Th42;
end;
