let G1 be _Graph; :: thesis: for G2 being Subgraph of G1
for E2 being RepDEdgeSelection of G2 ex E1 being RepDEdgeSelection of G1 st E2 = E1 /\ ()

let G2 be Subgraph of G1; :: thesis: for E2 being RepDEdgeSelection of G2 ex E1 being RepDEdgeSelection of G1 st E2 = E1 /\ ()
let E2 be RepDEdgeSelection of G2; :: thesis: ex E1 being RepDEdgeSelection of G1 st E2 = E1 /\ ()
set A = { { e where e is Element of the_Edges_of G1 : e DJoins v1,v2,G1 } where v1, v2 is Vertex of G1 : ( ex e0 being object st e0 DJoins v1,v2,G1 & ( for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 ) )
}
;
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 G1 : e DJoins v1,v2,G1 } where v1, v2 is Vertex of G1 : ( ex e0 being object st e0 DJoins v1,v2,G1 & ( for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 ) )
}
& 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 G1 : e DJoins v1,v2,G1 } where v1, v2 is Vertex of G1 : ( ex e0 being object st e0 DJoins v1,v2,G1 & ( for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 ) )
}
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 G1 : e DJoins v1,v2,G1 } where v1, v2 is Vertex of G1 : ( ex e0 being object st e0 DJoins v1,v2,G1 & ( for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 ) )
}
implies ex y being object st S1[x,y] )

assume x in { { e where e is Element of the_Edges_of G1 : e DJoins v1,v2,G1 } where v1, v2 is Vertex of G1 : ( ex e0 being object st e0 DJoins v1,v2,G1 & ( for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 ) )
}
; :: thesis: ex y being object st S1[x,y]
then consider v1, v2 being Vertex of G1 such that
A3: x = { e where e is Element of the_Edges_of G1 : e DJoins v1,v2,G1 } and
A4: ex e0 being object st e0 DJoins v1,v2,G1 and
for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 ;
reconsider B = x as set by A3;
consider e0 being object such that
A5: e0 DJoins v1,v2,G1 by A4;
reconsider e0 = e0 as Element of the_Edges_of G1 by ;
e0 in B by A3, A5;
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;
consider f being Function such that
A6: ( dom f = { { e where e is Element of the_Edges_of G1 : e DJoins v1,v2,G1 } where v1, v2 is Vertex of G1 : ( ex e0 being object st e0 DJoins v1,v2,G1 & ( for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 ) )
}
& ( for x being object st x in { { e where e is Element of the_Edges_of G1 : e DJoins v1,v2,G1 } where v1, v2 is Vertex of G1 : ( ex e0 being object st e0 DJoins v1,v2,G1 & ( for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 ) )
}
holds
S1[x,f . x] ) ) from for e being object st e in rng f holds
e in the_Edges_of G1
proof
let e be object ; :: thesis: ( e in rng f implies e in the_Edges_of G1 )
assume e in rng f ; :: thesis:
then consider C being object such that
A7: ( C in dom f & f . C = e ) by FUNCT_1:def 3;
consider C0 being non empty set such that
A8: ( C = C0 & f . C = the Element of C0 ) by A6, A7;
consider v1, v2 being Vertex of G1 such that
A9: C = { e where e is Element of the_Edges_of G1 : e DJoins v1,v2,G1 } and
ex e0 being object st e0 DJoins v1,v2,G1 and
for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 by A6, A7;
e in C0 by A7, A8;
then consider e2 being Element of the_Edges_of G1 such that
A10: ( e = e2 & e2 DJoins v1,v2,G1 ) by A8, A9;
thus e in the_Edges_of G1 by ; :: thesis: verum
end;
then A11: rng f c= the_Edges_of G1 by TARSKI:def 3;
the_Edges_of G2 c= the_Edges_of G1 ;
then E2 c= the_Edges_of G1 by XBOOLE_1:1;
then reconsider E1 = E2 \/ (rng f) as Subset of () by ;
for v, w, e0 being object st e0 DJoins v,w,G1 holds
ex e being object st
( e DJoins v,w,G1 & e in E1 & ( for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e ) )
proof
let v, w, e0 be object ; :: thesis: ( e0 DJoins v,w,G1 implies ex e being object st
( e DJoins v,w,G1 & e in E1 & ( for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e ) ) )

A12: ( v is set & w is set ) by TARSKI:1;
assume A13: e0 DJoins v,w,G1 ; :: thesis: ex e being object st
( e DJoins v,w,G1 & e in E1 & ( for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e ) )

per cases ( ex e1 being object st
( e1 DJoins v,w,G1 & e1 in E2 ) or for e1 being object st e1 DJoins v,w,G1 holds
not e1 in E2 )
;
suppose ex e1 being object st
( e1 DJoins v,w,G1 & e1 in E2 ) ; :: thesis: ex e being object st
( e DJoins v,w,G1 & e in E1 & ( for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e ) )

then consider e1 being object such that
A14: ( e1 DJoins v,w,G1 & e1 in E2 ) ;
e1 DJoins v,w,G2 by ;
then consider e being object such that
A15: ( e DJoins v,w,G2 & e in E2 ) and
A16: for e8 being object st e8 DJoins v,w,G2 & e8 in E2 holds
e8 = e by Def6;
take e ; :: thesis: ( e DJoins v,w,G1 & e in E1 & ( for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e ) )

thus A17: e DJoins v,w,G1 by ; :: thesis: ( e in E1 & ( for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e ) )

thus e in E1 by ; :: thesis: for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e

let e9 be object ; :: thesis: ( e9 DJoins v,w,G1 & e9 in E1 implies e9 = e )
assume A18: ( e9 DJoins v,w,G1 & e9 in E1 ) ; :: thesis: e9 = e
not e9 in rng f
proof
assume e9 in rng f ; :: thesis: contradiction
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 G1 such that
A20: C = { k where k is Element of the_Edges_of G1 : k DJoins v1,v2,G1 } and
ex k0 being object st k0 DJoins v1,v2,G1 and
A21: for k0 being object st k0 DJoins v1,v2,G1 holds
not k0 in E2 by ;
consider C0 being non empty set such that
A22: ( C = C0 & f . C = the Element of C0 ) by ;
e9 in C0 by ;
then consider k being Element of the_Edges_of G1 such that
A23: ( e9 = k & k DJoins v1,v2,G1 ) by ;
( v1 = v & v2 = w ) by ;
hence contradiction by A15, A17, A21; :: thesis: verum
end;
then A24: e9 in E2 by ;
then e9 DJoins v,w,G2 by ;
hence e9 = e by ; :: thesis: verum
end;
suppose A25: for e1 being object st e1 DJoins v,w,G1 holds
not e1 in E2 ; :: thesis: ex e being object st
( e DJoins v,w,G1 & e in E1 & ( for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e ) )

e0 Joins v,w,G1 by ;
then A26: ( v is Vertex of G1 & w is Vertex of G1 ) by GLIB_000:13;
set B = { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } ;
A27: { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } in { { e where e is Element of the_Edges_of G1 : e DJoins v1,v2,G1 } where v1, v2 is Vertex of G1 : ( ex e0 being object st e0 DJoins v1,v2,G1 & ( for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 ) )
}
by ;
then consider B0 being non empty set such that
A28: ( { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } = B0 & f . { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } = the Element of B0 ) by A6;
f . { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } in { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } by A28;
then consider e being Element of the_Edges_of G1 such that
A29: ( f . { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } = e & e DJoins v,w,G1 ) ;
take e ; :: thesis: ( e DJoins v,w,G1 & e in E1 & ( for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e ) )

thus e DJoins v,w,G1 by A29; :: thesis: ( e in E1 & ( for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e ) )

e in rng f by ;
hence e in E1 by XBOOLE_0:def 3; :: thesis: for e9 being object st e9 DJoins v,w,G1 & e9 in E1 holds
e9 = e

let e9 be object ; :: thesis: ( e9 DJoins v,w,G1 & e9 in E1 implies e9 = e )
assume A30: ( e9 DJoins v,w,G1 & e9 in E1 ) ; :: thesis: e9 = e
then not e9 in E2 by A25;
then e9 in rng f by ;
then consider C being object such that
A31: ( C in dom f & f . C = e9 ) by FUNCT_1:def 3;
consider v1, v2 being Vertex of G1 such that
A32: C = { k where k is Element of the_Edges_of G1 : k DJoins v1,v2,G1 } and
ex k0 being object st k0 DJoins v1,v2,G1 and
for k0 being object st k0 DJoins v1,v2,G1 holds
not k0 in E2 by ;
consider C0 being non empty set such that
A33: ( C = C0 & f . C = the Element of C0 ) by ;
f . C in C0 by A33;
then consider k being Element of the_Edges_of G1 such that
A34: ( f . C = k & k DJoins v1,v2,G1 ) by ;
for x being object holds
( x in { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } iff x in C0 )
proof
let x be object ; :: thesis: ( x in { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } iff x in C0 )
A35: ( v1 = v & v2 = w ) by A30, A31, A34, Th6;
thus ( x in { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } implies x in C0 ) by ; :: thesis: ( x in C0 implies x in { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } )
assume x in C0 ; :: thesis: x in { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 }
then consider k being Element of the_Edges_of G1 such that
A36: ( x = k & k DJoins v1,v2,G1 ) by ;
thus x in { e where e is Element of the_Edges_of G1 : e DJoins v,w,G1 } by ; :: thesis: verum
end;
hence e9 = e by ; :: thesis: verum
end;
end;
end;
then reconsider E1 = E1 as RepDEdgeSelection of G1 by Def6;
take E1 ; :: thesis: E2 = E1 /\ ()
for x being object holds
( x in E2 iff ( x in E1 & x in the_Edges_of G2 ) )
proof
let x be object ; :: thesis: ( x in E2 iff ( x in E1 & x in the_Edges_of G2 ) )
thus ( x in E2 implies ( x in E1 & x in the_Edges_of G2 ) ) by ; :: thesis: ( x in E1 & x in the_Edges_of G2 implies x in E2 )
assume A37: ( x in E1 & x in the_Edges_of G2 ) ; :: thesis: x in E2
not x in rng f
proof
assume x in rng f ; :: thesis: contradiction
then consider C being object such that
A38: ( C in dom f & f . C = x ) by FUNCT_1:def 3;
consider v1, v2 being Vertex of G1 such that
A39: C = { e where e is Element of the_Edges_of G1 : e DJoins v1,v2,G1 } and
ex e0 being object st e0 DJoins v1,v2,G1 and
A40: for e0 being object st e0 DJoins v1,v2,G1 holds
not e0 in E2 by ;
consider C0 being non empty set such that
A41: ( C = C0 & f . C = the Element of C0 ) by ;
f . C in C0 by A41;
then consider e being Element of the_Edges_of G1 such that
A42: ( f . C = e & e DJoins v1,v2,G1 ) by ;
x DJoins v1,v2,G2 by ;
then consider e1 being object such that
A43: ( e1 DJoins v1,v2,G2 & e1 in E2 ) and
for e9 being object st e9 DJoins v1,v2,G2 & e9 in E2 holds
e9 = e1 by Def6;
thus contradiction by A40, A43, GLIB_000:72; :: thesis: verum
end;
hence x in E2 by ; :: thesis: verum
end;
hence E2 = E1 /\ () by XBOOLE_0:def 4; :: thesis: verum