let G be _finite _Graph; :: thesis: for L being LexBFS:Labeling of G
for v being Vertex of G
for x being set
for k being Nat st not x in G .AdjacentSet {v} holds
(L `2) . x = ((LexBFS:Update (L,v,k)) `2) . x
let L be LexBFS:Labeling of G; :: thesis: for v being Vertex of G
for x being set
for k being Nat st not x in G .AdjacentSet {v} holds
(L `2) . x = ((LexBFS:Update (L,v,k)) `2) . x
let v be Vertex of G; :: thesis: for x being set
for k being Nat st not x in G .AdjacentSet {v} holds
(L `2) . x = ((LexBFS:Update (L,v,k)) `2) . x
let x be set ; :: thesis: for k being Nat st not x in G .AdjacentSet {v} holds
(L `2) . x = ((LexBFS:Update (L,v,k)) `2) . x
let k be Nat; :: thesis: ( not x in G .AdjacentSet {v} implies (L `2) . x = ((LexBFS:Update (L,v,k)) `2) . x )
assume A1: not x in G .AdjacentSet {v} ; :: thesis: (L `2) . x = ((LexBFS:Update (L,v,k)) `2) . x
set F = ((G .AdjacentSet {v}) \ (dom (L `1))) --> {((G .order()) -' k)};
A2: not x in dom (((G .AdjacentSet {v}) \ (dom (L `1))) --> {((G .order()) -' k)}) by A1, XBOOLE_0:def 5;
then A3: (((G .AdjacentSet {v}) \ (dom (L `1))) --> {((G .order()) -' k)}) . x = {} by FUNCT_1:def 2;
set L2 = (LexBFS:Update (L,v,k)) `2 ;

per cases ( x in dom (L `2) or not x in dom (L `2) ) ;

suppose x in dom (L `2) ; :: thesis: (L `2) . x = ((LexBFS:Update (L,v,k)) `2) . x

then x in (dom (L `2)) \/ (dom (((G .AdjacentSet {v}) \ (dom (L `1))) --> {((G .order()) -' k)})) by XBOOLE_0:def 3;
hence ((LexBFS:Update (L,v,k)) `2) . x = ((L `2) . x) \/ ((((G .AdjacentSet {v}) \ (dom (L `1))) --> {((G .order()) -' k)}) . x) by Def1
.= (L `2) . x by A3 ;
:: thesis: verum

end;

suppose A4: not x in dom (L `2) ; :: thesis: (L `2) . x = ((LexBFS:Update (L,v,k)) `2) . x

then not x in (dom (L `2)) \/ (dom (((G .AdjacentSet {v}) \ (dom (L `1))) --> {((G .order()) -' k)})) by A2, XBOOLE_0:def 3;
then not x in dom ((LexBFS:Update (L,v,k)) `2) by Def1;
hence ((LexBFS:Update (L,v,k)) `2) . x = {} by FUNCT_1:def 2
.= (L `2) . x by A4, FUNCT_1:def 2 ;
:: thesis: verum

end;

end;