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 x in dom (L `1 ) holds
((LexBFS:Update L,v,k) `2 ) . x = (L `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 x in dom (L `1 ) holds
((LexBFS:Update L,v,k) `2 ) . x = (L `2 ) . x
let v be Vertex of G; :: thesis: for x being set
for k being Nat st x in dom (L `1 ) holds
((LexBFS:Update L,v,k) `2 ) . x = (L `2 ) . x
let x be set ; :: thesis: for k being Nat st x in dom (L `1 ) holds
((LexBFS:Update L,v,k) `2 ) . x = (L `2 ) . x
let k be Nat; :: thesis: ( x in dom (L `1 ) implies ((LexBFS:Update L,v,k) `2 ) . x = (L `2 ) . x )
assume A: x in dom (L `1 ) ; :: thesis: ((LexBFS:Update L,v,k) `2 ) . x = (L `2 ) . x
set L2 = (LexBFS:Update L,v,k) `2 ;
set F = ((G .AdjacentSet {v}) \ (dom (L `1 ))) --> {((G .order() ) -' k)};
C: (LexBFS:Update L,v,k) `2 = (L `2 ) .\/ (((G .AdjacentSet {v}) \ (dom (L `1 ))) --> {((G .order() ) -' k)}) by MCART_1:7;
D: dom (((G .AdjacentSet {v}) \ (dom (L `1 ))) --> {((G .order() ) -' k)}) = (G .AdjacentSet {v}) \ (dom (L `1 )) by FUNCOP_1:19;
E: not x in dom (((G .AdjacentSet {v}) \ (dom (L `1 ))) --> {((G .order() ) -' k)}) by A, D, XBOOLE_0:def 5;
F: (((G .AdjacentSet {v}) \ (dom (L `1 ))) --> {((G .order() ) -' k)}) . x = {} by E, FUNCT_1:def 4;

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

suppose x in dom (L `2 ) ; :: thesis: ((LexBFS:Update L,v,k) `2 ) . x = (L `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 C, Def2
.= (L `2 ) . x by F ;
:: thesis: verum

end;

suppose S1: not x in dom (L `2 ) ; :: thesis: ((LexBFS:Update L,v,k) `2 ) . x = (L `2 ) . x

then not x in (dom (L `2 )) \/ (dom (((G .AdjacentSet {v}) \ (dom (L `1 ))) --> {((G .order() ) -' k)})) by E, XBOOLE_0:def 3;
then not x in dom ((LexBFS:Update L,v,k) `2 ) by C, Def2;
hence ((LexBFS:Update L,v,k) `2 ) . x = {} by FUNCT_1:def 4
.= (L `2 ) . x by S1, FUNCT_1:def 4 ;
:: thesis: verum

end;

end;