
theorem Th36:
  for T being _Tree for a, b, c being Vertex of T holds c in T
.pathBetween(a,b).vertices() iff T.pathBetween(a,b) = T.pathBetween(a,c).append
  (T.pathBetween(c,b))
proof
  let T be _Tree, a, b, c be Vertex of T;
  set P = T.pathBetween(a,b);
  set ci = P.find(c);
  set pac = T.pathBetween(a,c), pcb = T.pathBetween(c,b);
  hereby
A1: P = P.cut(1, len P) by GLIB_001:39;
A2: 1 <= ci & 1 = 2*0+1 by ABIAN:12;
    assume
A3: c in P.vertices();
    then
A4: ci <= len P by GLIB_001:def 19;
A5: P.ci = c by A3,GLIB_001:def 19;
    P.len P = P.last() .= b by Th28;
    then
A6: pcb = P.cut(ci, len P) by A4,A5,Th35;
    P.1 = P.first() .= a by Th28;
    then pac = P.cut(1,ci) by A4,A2,A5,Th35;
    hence P = T.pathBetween(a,c).append(T.pathBetween(c,b)) by A4,A2,A6,A1,
GLIB_001:38;
  end;
  assume P = pac.append(pcb);
  then
A7: pac.vertices() c= P.vertices() by Th13,Th15;
  c in pac.vertices() by Th29;
  hence thesis by A7;
end;
