let c be Cardinal; :: thesis: for G2 being non edgeless _Graph
for v, e, w being object
for G1 being addAdjVertex of G2,v,e,w st G2 is c -tcolorable holds
G1 is c +` 1 -tcolorable
let G2 be non edgeless _Graph; :: thesis: for v, e, w being object
for G1 being addAdjVertex of G2,v,e,w st G2 is c -tcolorable holds
G1 is c +` 1 -tcolorable
let v, e, w be object ; :: thesis: for G1 being addAdjVertex of G2,v,e,w st G2 is c -tcolorable holds
G1 is c +` 1 -tcolorable
let G1 be addAdjVertex of G2,v,e,w; :: thesis: ( G2 is c -tcolorable implies G1 is c +` 1 -tcolorable )
assume A1: G2 is c -tcolorable ; :: thesis: G1 is c +` 1 -tcolorable

per cases ( ( not e in the_Edges_of G2 & not v in the_Vertices_of G2 & w in the_Vertices_of G2 ) or ( not e in the_Edges_of G2 & v in the_Vertices_of G2 & not w in the_Vertices_of G2 ) or ( not ( not e in the_Edges_of G2 & not v in the_Vertices_of G2 & w in the_Vertices_of G2 ) & not ( not e in the_Edges_of G2 & v in the_Vertices_of G2 & not w in the_Vertices_of G2 ) ) ) ;

suppose ( not e in the_Edges_of G2 & not v in the_Vertices_of G2 & w in the_Vertices_of G2 ) ; :: thesis: G1 is c +` 1 -tcolorable

hence G1 is c +` 1 -tcolorable by A1, Lm18; :: thesis: verum

end;

suppose A2: ( not e in the_Edges_of G2 & v in the_Vertices_of G2 & not w in the_Vertices_of G2 ) ; :: thesis: G1 is c +` 1 -tcolorable

set G3 = the reverseEdgeDirections of G1,{e};
the reverseEdgeDirections of G1,{e} is addAdjVertex of G2,w,e,v by A2, GLIBPRE1:66;
then the reverseEdgeDirections of G1,{e} is c +` 1 -tcolorable by A1, A2, Lm18;
hence G1 is c +` 1 -tcolorable by Th168; :: thesis: verum

end;

suppose ( not ( not e in the_Edges_of G2 & not v in the_Vertices_of G2 & w in the_Vertices_of G2 ) & not ( not e in the_Edges_of G2 & v in the_Vertices_of G2 & not w in the_Vertices_of G2 ) ) ; :: thesis: G1 is c +` 1 -tcolorable

then G1 == G2 by GLIB_006:def 12;
then G1 is c -tcolorable by A1, Th167;
hence G1 is c +` 1 -tcolorable by Th161, CARD_2:94; :: thesis: verum

end;

end;