let G2 be _Graph; :: thesis: for v being object
for V being finite set
for G1 being addAdjVertexAll of G2,v,V holds
( G1 is finite-tcolorable iff G2 is finite-tcolorable )
let v be object ; :: thesis: for V being finite set
for G1 being addAdjVertexAll of G2,v,V holds
( G1 is finite-tcolorable iff G2 is finite-tcolorable )
let V be finite set ; :: thesis: for G1 being addAdjVertexAll of G2,v,V holds
( G1 is finite-tcolorable iff G2 is finite-tcolorable )
let G1 be addAdjVertexAll of G2,v,V; :: 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 ;
G1 is (n +` 1) +` (card V) -tcolorable by A2, Th175;
hence G1 is finite-tcolorable ; :: thesis: verum