let G2 be edgeless _Graph; :: thesis: for v, e, w being object
for G1 being addAdjVertex of G2,v,e,w holds G1 is 1 -ecolorable
let v, e, w be object ; :: thesis: for G1 being addAdjVertex of G2,v,e,w holds G1 is 1 -ecolorable
let G1 be addAdjVertex of G2,v,e,w; :: thesis: G1 is 1 -ecolorable

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 & v in the_Vertices_of G2 & not w in the_Vertices_of G2 ) & not ( not e in the_Edges_of G2 & not v in the_Vertices_of G2 & 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 1 -ecolorable

hence G1 is 1 -ecolorable by Lm13; :: thesis: verum

end;

suppose A1: ( not e in the_Edges_of G2 & v in the_Vertices_of G2 & not w in the_Vertices_of G2 ) ; :: thesis: G1 is 1 -ecolorable

set G3 = the reverseEdgeDirections of G1,{e};
the reverseEdgeDirections of G1,{e} is addAdjVertex of G2,w,e,v by A1, GLIBPRE1:66;
then the reverseEdgeDirections of G1,{e} is 1 -ecolorable by A1, Lm13;
hence G1 is 1 -ecolorable by Th104; :: thesis: verum

end;

suppose ( not ( not e in the_Edges_of G2 & v in the_Vertices_of G2 & not w in the_Vertices_of G2 ) & not ( not e in the_Edges_of G2 & not v in the_Vertices_of G2 & w in the_Vertices_of G2 ) ) ; :: thesis: G1 is 1 -ecolorable

then G1 == G2 by GLIB_006:def 12;
then G1 is edgeless by GLIB_008:52;
hence G1 is 1 -ecolorable by Th99, XBOOLE_1:2; :: thesis: verum

end;

end;