let G be _Graph; :: thesis: for E1 being RepDEdgeSelection of G ex E2 being RepEdgeSelection of G st E2 c= E1
let E1 be RepDEdgeSelection of G; :: thesis: ex E2 being RepEdgeSelection of G st E2 c= E1
set A = { { e where e is Element of the_Edges_of G : ( e Joins v1,v2,G & e in E1 ) } where v1, v2 is Vertex of G : ex e0 being object st e0 Joins v1,v2,G } ;
defpred S₁[ object , object ] means ex S being non empty set st
( $1 = S & $2 = the Element of S );
A1: for x, y1, y2 being object st x in { { e where e is Element of the_Edges_of G : ( e Joins v1,v2,G & e in E1 ) } where v1, v2 is Vertex of G : ex e0 being object st e0 Joins v1,v2,G } & S₁[x,y1] & S₁[x,y2] holds
y1 = y2 ;
A2: for x being object st x in { { e where e is Element of the_Edges_of G : ( e Joins v1,v2,G & e in E1 ) } where v1, v2 is Vertex of G : ex e0 being object st e0 Joins v1,v2,G } holds
ex y being object st S₁[x,y]

proof

let x be object ; :: thesis: ( x in { { e where e is Element of the_Edges_of G : ( e Joins v1,v2,G & e in E1 ) } where v1, v2 is Vertex of G : ex e0 being object st e0 Joins v1,v2,G } implies ex y being object st S₁[x,y] )
assume x in { { e where e is Element of the_Edges_of G : ( e Joins v1,v2,G & e in E1 ) } where v1, v2 is Vertex of G : ex e0 being object st e0 Joins v1,v2,G } ; :: thesis: ex y being object st S₁[x,y]
then consider v1, v2 being Vertex of G such that
A3: x = { e where e is Element of the_Edges_of G : ( e Joins v1,v2,G & e in E1 ) } and
A4: ex e0 being object st e0 Joins v1,v2,G ;
reconsider B = x as set by A3;
consider e0 being object such that
A5: e0 Joins v1,v2,G by A4;

per cases ( e0 DJoins v1,v2,G or e0 DJoins v2,v1,G ) by A5, GLIB_000:16;

suppose e0 DJoins v1,v2,G ; :: thesis: ex y being object st S₁[x,y]

then consider e being object such that
A6: ( e DJoins v1,v2,G & e in E1 ) and
for e9 being object st e9 DJoins v1,v2,G & e9 in E1 holds
e9 = e by Def6;
( e in the_Edges_of G & e Joins v1,v2,G ) by A6, GLIB_000:16;
then e in B by A3, A6;
then reconsider B = B as non empty set ;
take the Element of B ; :: thesis: S₁[x, the Element of B]
take B ; :: thesis: ( x = B & the Element of B = the Element of B )
thus ( x = B & the Element of B = the Element of B ) ; :: thesis: verum

end;

suppose e0 DJoins v2,v1,G ; :: thesis: ex y being object st S₁[x,y]

then consider e being object such that
A7: ( e DJoins v2,v1,G & e in E1 ) and
for e9 being object st e9 DJoins v2,v1,G & e9 in E1 holds
e9 = e by Def6;
( e in the_Edges_of G & e Joins v1,v2,G ) by A7, GLIB_000:16;
then e in B by A3, A7;
then reconsider B = B as non empty set ;
take the Element of B ; :: thesis: S₁[x, the Element of B]
take B ; :: thesis: ( x = B & the Element of B = the Element of B )
thus ( x = B & the Element of B = the Element of B ) ; :: thesis: verum

end;

end;

end;

consider f being Function such that
A8: ( dom f = { { e where e is Element of the_Edges_of G : ( e Joins v1,v2,G & e in E1 ) } where v1, v2 is Vertex of G : ex e0 being object st e0 Joins v1,v2,G } & ( for x being object st x in { { e where e is Element of the_Edges_of G : ( e Joins v1,v2,G & e in E1 ) } where v1, v2 is Vertex of G : ex e0 being object st e0 Joins v1,v2,G } holds
S₁[x,f . x] ) ) from FUNCT_1:sch 2(A1, A2);
for e being object st e in rng f holds
e in E1

proof

let e be object ; :: thesis: ( e in rng f implies e in E1 )
assume e in rng f ; :: thesis: e in E1
then consider C being object such that
A9: ( C in dom f & f . C = e ) by FUNCT_1:def 3;
consider C0 being non empty set such that
A10: ( C = C0 & f . C = the Element of C0 ) by A8, A9;
consider v1, v2 being Vertex of G such that
A11: C = { e2 where e2 is Element of the_Edges_of G : ( e2 Joins v1,v2,G & e2 in E1 ) } and
ex e8 being object st e8 Joins v1,v2,G by A8, A9;
e in C0 by A9, A10;
then consider e2 being Element of the_Edges_of G such that
A12: ( e = e2 & e2 Joins v1,v2,G & e2 in E1 ) by A10, A11;
thus e in E1 by A12; :: thesis: verum

end;

then A13: rng f c= E1 by TARSKI:def 3;
then reconsider E2 = rng f as Subset of (the_Edges_of G) by XBOOLE_1:1;
for v, w, e0 being object st e0 Joins v,w,G holds
ex e being object st
( e Joins v,w,G & e in E2 & ( for e9 being object st e9 Joins v,w,G & e9 in E2 holds
e9 = e ) )

proof

let v, w, e0 be object ; :: thesis: ( e0 Joins v,w,G implies ex e being object st
( e Joins v,w,G & e in E2 & ( for e9 being object st e9 Joins v,w,G & e9 in E2 holds
e9 = e ) ) )
assume A14: e0 Joins v,w,G ; :: thesis: ex e being object st
( e Joins v,w,G & e in E2 & ( for e9 being object st e9 Joins v,w,G & e9 in E2 holds
e9 = e ) )
set B = { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } ;
( v in the_Vertices_of G & w in the_Vertices_of G ) by A14, GLIB_000:13;
then A15: { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } in { { e where e is Element of the_Edges_of G : ( e Joins v1,v2,G & e in E1 ) } where v1, v2 is Vertex of G : ex e0 being object st e0 Joins v1,v2,G } by A14;
then consider B0 being non empty set such that
A16: ( { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } = B0 & f . { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } = the Element of B0 ) by A8;
f . { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } in { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } by A16;
then consider e being Element of the_Edges_of G such that
A17: ( f . { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } = e & e Joins v,w,G & e in E1 ) ;
take e ; :: thesis: ( e Joins v,w,G & e in E2 & ( for e9 being object st e9 Joins v,w,G & e9 in E2 holds
e9 = e ) )
thus e Joins v,w,G by A17; :: thesis: ( e in E2 & ( for e9 being object st e9 Joins v,w,G & e9 in E2 holds
e9 = e ) )
thus e in E2 by A8, A15, A17, FUNCT_1:3; :: thesis: for e9 being object st e9 Joins v,w,G & e9 in E2 holds
e9 = e
let e9 be object ; :: thesis: ( e9 Joins v,w,G & e9 in E2 implies e9 = e )
assume A18: ( e9 Joins v,w,G & e9 in E2 ) ; :: thesis: e9 = e
then consider C being object such that
A19: ( C in dom f & f . C = e9 ) by FUNCT_1:def 3;
consider v1, v2 being Vertex of G such that
A20: C = { e2 where e2 is Element of the_Edges_of G : ( e2 Joins v1,v2,G & e2 in E1 ) } and
ex e8 being object st e8 Joins v1,v2,G by A8, A19;
consider C0 being non empty set such that
A21: ( C = C0 & f . C = the Element of C0 ) by A8, A19;
e9 in C0 by A19, A21;
then consider e2 being Element of the_Edges_of G such that
A22: ( e9 = e2 & e2 Joins v1,v2,G & e2 in E1 ) by A20, A21;

per cases ( ( v = v1 & w = v2 ) or ( v = v2 & w = v1 ) ) by A18, A22, GLIB_000:15;

suppose ( v = v1 & w = v2 ) ; :: thesis: e9 = e

hence e9 = e by A17, A19, A20; :: thesis: verum

end;

suppose A23: ( v = v2 & w = v1 ) ; :: thesis: e9 = e

for x being object holds
( x in C0 iff x in { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } )

proof

let x be object ; :: thesis: ( x in C0 iff x in { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } )

hereby :: thesis: ( x in { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } implies x in C0 )

assume x in C0 ; :: thesis: x in { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) }
then consider e2 being Element of the_Edges_of G such that
A24: ( x = e2 & e2 Joins v1,v2,G & e2 in E1 ) by A20, A21;
e2 Joins v,w,G by A23, A24, GLIB_000:14;
hence x in { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } by A24; :: thesis: verum

end;

assume x in { e where e is Element of the_Edges_of G : ( e Joins v,w,G & e in E1 ) } ; :: thesis: x in C0
then consider e1 being Element of the_Edges_of G such that
A25: ( x = e1 & e1 Joins v,w,G & e1 in E1 ) ;
e1 Joins v1,v2,G by A23, A25, GLIB_000:14;
hence x in C0 by A20, A21, A25; :: thesis: verum

end;

hence e9 = e by A17, A19, A21, TARSKI:2; :: thesis: verum

end;

end;

end;

then reconsider E2 = E2 as RepEdgeSelection of G by Def5;
take E2 ; :: thesis: E2 c= E1
thus E2 c= E1 by A13; :: thesis: verum