let G1, G2 be _Graph; :: thesis: for F being PGraphMapping of G1,G2
for X being Subset of () st F is weak_SG-embedding holds
card () c= card (G2 .edgesBetween ((F _V) .: X))

let F be PGraphMapping of G1,G2; :: thesis: for X being Subset of () st F is weak_SG-embedding holds
card () c= card (G2 .edgesBetween ((F _V) .: X))

let X be Subset of (); :: thesis: ( F is weak_SG-embedding implies card () c= card (G2 .edgesBetween ((F _V) .: X)) )
assume A1: F is weak_SG-embedding ; :: thesis: card () c= card (G2 .edgesBetween ((F _V) .: X))
set f = (F _E) | ();
A2: dom ((F _E) | ()) = (dom (F _E)) /\ () by RELAT_1:61
.= () /\ () by
.= G1 .edgesBetween X by XBOOLE_1:28 ;
for y being object st y in rng ((F _E) | ()) holds
y in G2 .edgesBetween ((F _V) .: X)
proof
let y be object ; :: thesis: ( y in rng ((F _E) | ()) implies y in G2 .edgesBetween ((F _V) .: X) )
assume y in rng ((F _E) | ()) ; :: thesis: y in G2 .edgesBetween ((F _V) .: X)
then consider x being object such that
A3: ( x in dom ((F _E) | ()) & ((F _E) | ()) . x = y ) by FUNCT_1:def 3;
set v = () . x;
set w = () . x;
A4: ( x in the_Edges_of G1 & () . x in X & () . x in X ) by ;
then ( (the_Source_of G1) . x in the_Vertices_of G1 & () . x in the_Vertices_of G1 ) ;
then A5: ( (the_Source_of G1) . x in dom (F _V) & () . x in dom (F _V) ) by ;
A6: x in dom (F _E) by ;
x Joins () . x,() . x,G1 by ;
then (F _E) . x Joins (F _V) . (() . x),(F _V) . (() . x),G2 by A5, A6, Th4;
then A7: y Joins (F _V) . (() . x),(F _V) . (() . x),G2 by ;
( (F _V) . (() . x) in (F _V) .: X & (F _V) . (() . x) in (F _V) .: X ) by ;
hence y in G2 .edgesBetween ((F _V) .: X) by ; :: thesis: verum
end;
then A8: rng ((F _E) | ()) c= G2 .edgesBetween ((F _V) .: X) by TARSKI:def 3;
(F _E) | () is one-to-one by ;
hence card () c= card (G2 .edgesBetween ((F _V) .: X)) by ; :: thesis: verum