let G1 be non _trivial _Graph; :: thesis: for v being Vertex of G1
for G2 being removeVertex of G1,v
for v1 being Vertex of G1
for v2 being Vertex of G2 st v1 = v2 holds
( v2 .edgesIn() = (v1 .edgesIn()) \ (v .edgesOut()) & v2 .edgesOut() = (v1 .edgesOut()) \ (v .edgesIn()) & v2 .edgesInOut() = (v1 .edgesInOut()) \ (v .edgesInOut()) )
let v be Vertex of G1; :: thesis: for G2 being removeVertex of G1,v
for v1 being Vertex of G1
for v2 being Vertex of G2 st v1 = v2 holds
( v2 .edgesIn() = (v1 .edgesIn()) \ (v .edgesOut()) & v2 .edgesOut() = (v1 .edgesOut()) \ (v .edgesIn()) & v2 .edgesInOut() = (v1 .edgesInOut()) \ (v .edgesInOut()) )
let G2 be removeVertex of G1,v; :: thesis: for v1 being Vertex of G1
for v2 being Vertex of G2 st v1 = v2 holds
( v2 .edgesIn() = (v1 .edgesIn()) \ (v .edgesOut()) & v2 .edgesOut() = (v1 .edgesOut()) \ (v .edgesIn()) & v2 .edgesInOut() = (v1 .edgesInOut()) \ (v .edgesInOut()) )
let v1 be Vertex of G1; :: thesis: for v2 being Vertex of G2 st v1 = v2 holds
( v2 .edgesIn() = (v1 .edgesIn()) \ (v .edgesOut()) & v2 .edgesOut() = (v1 .edgesOut()) \ (v .edgesIn()) & v2 .edgesInOut() = (v1 .edgesInOut()) \ (v .edgesInOut()) )
let v2 be Vertex of G2; :: thesis: ( v1 = v2 implies ( v2 .edgesIn() = (v1 .edgesIn()) \ (v .edgesOut()) & v2 .edgesOut() = (v1 .edgesOut()) \ (v .edgesIn()) & v2 .edgesInOut() = (v1 .edgesInOut()) \ (v .edgesInOut()) ) )
assume A1: v1 = v2 ; :: thesis: ( v2 .edgesIn() = (v1 .edgesIn()) \ (v .edgesOut()) & v2 .edgesOut() = (v1 .edgesOut()) \ (v .edgesIn()) & v2 .edgesInOut() = (v1 .edgesInOut()) \ (v .edgesInOut()) )
(the_Vertices_of G1) \ {v} <> {} by GLIB_000:20;
then the_Vertices_of G1 <> {v} by XBOOLE_1:37;
then {v} c< the_Vertices_of G1 by XBOOLE_0:def 8;
hence ( v2 .edgesIn() = (v1 .edgesIn()) \ (v .edgesOut()) & v2 .edgesOut() = (v1 .edgesOut()) \ (v .edgesIn()) & v2 .edgesInOut() = (v1 .edgesInOut()) \ (v .edgesInOut()) ) by A1, Th48; :: thesis: verum