let G be _finite _Graph; :: thesis: for L being MCS:Labeling of G

for v, x being set st x in dom (L `1) holds

(L `2) . x = ((MCS:LabelAdjacent (L,v)) `2) . x

let L be MCS:Labeling of G; :: thesis: for v, x being set st x in dom (L `1) holds

(L `2) . x = ((MCS:LabelAdjacent (L,v)) `2) . x

let v, x be set ; :: thesis: ( x in dom (L `1) implies (L `2) . x = ((MCS:LabelAdjacent (L,v)) `2) . x )

assume A1: x in dom (L `1) ; :: thesis: (L `2) . x = ((MCS:LabelAdjacent (L,v)) `2) . x

set V2G = L `2 ;

set VLG = L `1 ;

set GL = MCS:LabelAdjacent (L,v);

set V2 = (MCS:LabelAdjacent (L,v)) `2 ;

not x in (G .AdjacentSet {v}) \ (dom (L `1)) by A1, XBOOLE_0:def 5;

hence (L `2) . x = ((MCS:LabelAdjacent (L,v)) `2) . x by Def3; :: thesis: verum

for v, x being set st x in dom (L `1) holds

(L `2) . x = ((MCS:LabelAdjacent (L,v)) `2) . x

let L be MCS:Labeling of G; :: thesis: for v, x being set st x in dom (L `1) holds

(L `2) . x = ((MCS:LabelAdjacent (L,v)) `2) . x

let v, x be set ; :: thesis: ( x in dom (L `1) implies (L `2) . x = ((MCS:LabelAdjacent (L,v)) `2) . x )

assume A1: x in dom (L `1) ; :: thesis: (L `2) . x = ((MCS:LabelAdjacent (L,v)) `2) . x

set V2G = L `2 ;

set VLG = L `1 ;

set GL = MCS:LabelAdjacent (L,v);

set V2 = (MCS:LabelAdjacent (L,v)) `2 ;

not x in (G .AdjacentSet {v}) \ (dom (L `1)) by A1, XBOOLE_0:def 5;

hence (L `2) . x = ((MCS:LabelAdjacent (L,v)) `2) . x by Def3; :: thesis: verum