let G2 be _Graph; :: thesis: for v, e, w being object
for G1 being addAdjVertex of G2,v,e,w holds
( G1 is finite-tcolorable iff G2 is finite-tcolorable )
let v, e, w be object ; :: thesis: for G1 being addAdjVertex of G2,v,e,w holds
( G1 is finite-tcolorable iff G2 is finite-tcolorable )
let G1 be addAdjVertex of G2,v,e,w; :: thesis: ( G1 is finite-tcolorable iff G2 is finite-tcolorable )

hereby :: thesis: ( G2 is finite-tcolorable implies G1 is finite-tcolorable )

assume A1: G1 is finite-tcolorable ; :: thesis: G2 is finite-tcolorable
G2 is Subgraph of G1 by GLIB_006:57;
hence G2 is finite-tcolorable by A1; :: thesis: verum

end;

assume G2 is finite-tcolorable ; :: thesis: G1 is finite-tcolorable
then consider n being Nat such that
A2: G2 is n -tcolorable ;

per cases ( not G2 is edgeless or G2 is edgeless ) ;

suppose not G2 is edgeless ; :: thesis: G1 is finite-tcolorable

then G1 is n +` 1 -tcolorable by A2, Th172;
hence G1 is finite-tcolorable ; :: thesis: verum

end;

suppose G2 is edgeless ; :: thesis: G1 is finite-tcolorable

then G1 is 3 -tcolorable by Th174;
hence G1 is finite-tcolorable ; :: thesis: verum

end;

end;