let G1 be _Graph; :: thesis: for G2 being Subgraph of G1
for E being RepEdgeSelection of G1 st E c= the_Edges_of G2 holds
E is RepEdgeSelection of G2
let G2 be Subgraph of G1; :: thesis: for E being RepEdgeSelection of G1 st E c= the_Edges_of G2 holds
E is RepEdgeSelection of G2
let E be RepEdgeSelection of G1; :: thesis: ( E c= the_Edges_of G2 implies E is RepEdgeSelection of G2 )
assume A1: E c= the_Edges_of G2 ; :: thesis: E is RepEdgeSelection of G2
now :: thesis: for v, w, e0 being object st e0 Joins v,w,G2 holds
ex e being object st
( e Joins v,w,G2 & e in E & ( for e9 being object st e9 Joins v,w,G2 & e9 in E holds
e9 = e ) )
let v, w, e0 be object ; :: thesis: ( e0 Joins v,w,G2 implies ex e being object st
( e Joins v,w,G2 & e in E & ( for e9 being object st e9 Joins v,w,G2 & e9 in E holds
e9 = e ) ) )
A2: ( v is set & w is set ) by TARSKI:1;
assume e0 Joins v,w,G2 ; :: thesis: ex e being object st
( e Joins v,w,G2 & e in E & ( for e9 being object st e9 Joins v,w,G2 & e9 in E holds
e9 = e ) )
then consider e being object such that
A3: ( e Joins v,w,G1 & e in E ) and
A4: for e9 being object st e9 Joins v,w,G1 & e9 in E holds
e9 = e by A2, Def5, GLIB_000:72;
take e = e; :: thesis: ( e Joins v,w,G2 & e in E & ( for e9 being object st e9 Joins v,w,G2 & e9 in E holds
e9 = e ) )
thus ( e Joins v,w,G2 & e in E ) by A1, A2, A3, GLIB_000:73; :: thesis: for e9 being object st e9 Joins v,w,G2 & e9 in E holds
e9 = e
let e9 be object ; :: thesis: ( e9 Joins v,w,G2 & e9 in E implies e9 = e )
assume ( e9 Joins v,w,G2 & e9 in E ) ; :: thesis: e9 = e
hence e9 = e by A2, A4, GLIB_000:72; :: thesis: verum
end;
hence E is RepEdgeSelection of G2 by A1, Def5; :: thesis: verum