let G1 be _Graph; :: thesis: for G2 being removeDParallelEdges of G1
for v1 being Vertex of G1
for v2 being Vertex of G2 st v1 = v2 holds
( v1 is isolated iff v2 is isolated )
let G2 be removeDParallelEdges of G1; :: thesis: for v1 being Vertex of G1
for v2 being Vertex of G2 st v1 = v2 holds
( v1 is isolated iff v2 is isolated )
set G3 = the removeParallelEdges of G2;
let v1 be Vertex of G1; :: thesis: for v2 being Vertex of G2 st v1 = v2 holds
( v1 is isolated iff v2 is isolated )
let v2 be Vertex of G2; :: thesis: ( v1 = v2 implies ( v1 is isolated iff v2 is isolated ) )
assume A1: v1 = v2 ; :: thesis: ( v1 is isolated iff v2 is isolated )
reconsider v3 = v2 as Vertex of the removeParallelEdges of G2 by GLIB_000:def 33;
the removeParallelEdges of G2 is removeParallelEdges of G1 by Th95;
then ( v1 is isolated iff v3 is isolated ) by A1, Th111;
hence ( v1 is isolated iff v2 is isolated ) by Th111; :: thesis: verum