let G1, G2 be _Graph; :: thesis: for F being PGraphMapping of G1,G2

for X being Subset of (the_Vertices_of G1) st F is weak_SG-embedding holds

card (G1 .edgesBetween X) c= card (G2 .edgesBetween ((F _V) .: X))

let F be PGraphMapping of G1,G2; :: thesis: for X being Subset of (the_Vertices_of G1) st F is weak_SG-embedding holds

card (G1 .edgesBetween X) c= card (G2 .edgesBetween ((F _V) .: X))

let X be Subset of (the_Vertices_of G1); :: thesis: ( F is weak_SG-embedding implies card (G1 .edgesBetween X) c= card (G2 .edgesBetween ((F _V) .: X)) )

assume A1: F is weak_SG-embedding ; :: thesis: card (G1 .edgesBetween X) c= card (G2 .edgesBetween ((F _V) .: X))

set f = (F _E) | (G1 .edgesBetween X);

A2: dom ((F _E) | (G1 .edgesBetween X)) = (dom (F _E)) /\ (G1 .edgesBetween X) by RELAT_1:61

.= (the_Edges_of G1) /\ (G1 .edgesBetween X) by A1, Def11

.= G1 .edgesBetween X by XBOOLE_1:28 ;

for y being object st y in rng ((F _E) | (G1 .edgesBetween X)) holds

y in G2 .edgesBetween ((F _V) .: X)

(F _E) | (G1 .edgesBetween X) is one-to-one by A1, FUNCT_1:52;

hence card (G1 .edgesBetween X) c= card (G2 .edgesBetween ((F _V) .: X)) by A2, A8, CARD_1:10; :: thesis: verum

for X being Subset of (the_Vertices_of G1) st F is weak_SG-embedding holds

card (G1 .edgesBetween X) c= card (G2 .edgesBetween ((F _V) .: X))

let F be PGraphMapping of G1,G2; :: thesis: for X being Subset of (the_Vertices_of G1) st F is weak_SG-embedding holds

card (G1 .edgesBetween X) c= card (G2 .edgesBetween ((F _V) .: X))

let X be Subset of (the_Vertices_of G1); :: thesis: ( F is weak_SG-embedding implies card (G1 .edgesBetween X) c= card (G2 .edgesBetween ((F _V) .: X)) )

assume A1: F is weak_SG-embedding ; :: thesis: card (G1 .edgesBetween X) c= card (G2 .edgesBetween ((F _V) .: X))

set f = (F _E) | (G1 .edgesBetween X);

A2: dom ((F _E) | (G1 .edgesBetween X)) = (dom (F _E)) /\ (G1 .edgesBetween X) by RELAT_1:61

.= (the_Edges_of G1) /\ (G1 .edgesBetween X) by A1, Def11

.= G1 .edgesBetween X by XBOOLE_1:28 ;

for y being object st y in rng ((F _E) | (G1 .edgesBetween X)) holds

y in G2 .edgesBetween ((F _V) .: X)

proof

then A8:
rng ((F _E) | (G1 .edgesBetween X)) c= G2 .edgesBetween ((F _V) .: X)
by TARSKI:def 3;
let y be object ; :: thesis: ( y in rng ((F _E) | (G1 .edgesBetween X)) implies y in G2 .edgesBetween ((F _V) .: X) )

assume y in rng ((F _E) | (G1 .edgesBetween X)) ; :: thesis: y in G2 .edgesBetween ((F _V) .: X)

then consider x being object such that

A3: ( x in dom ((F _E) | (G1 .edgesBetween X)) & ((F _E) | (G1 .edgesBetween X)) . x = y ) by FUNCT_1:def 3;

set v = (the_Source_of G1) . x;

set w = (the_Target_of G1) . x;

A4: ( x in the_Edges_of G1 & (the_Source_of G1) . x in X & (the_Target_of G1) . x in X ) by A2, A3, GLIB_000:31;

then ( (the_Source_of G1) . x in the_Vertices_of G1 & (the_Target_of G1) . x in the_Vertices_of G1 ) ;

then A5: ( (the_Source_of G1) . x in dom (F _V) & (the_Target_of G1) . x in dom (F _V) ) by A1, Def11;

A6: x in dom (F _E) by A1, A4, Def11;

x Joins (the_Source_of G1) . x,(the_Target_of G1) . x,G1 by A4, GLIB_000:def 13;

then (F _E) . x Joins (F _V) . ((the_Source_of G1) . x),(F _V) . ((the_Target_of G1) . x),G2 by A5, A6, Th4;

then A7: y Joins (F _V) . ((the_Source_of G1) . x),(F _V) . ((the_Target_of G1) . x),G2 by A3, FUNCT_1:47;

( (F _V) . ((the_Source_of G1) . x) in (F _V) .: X & (F _V) . ((the_Target_of G1) . x) in (F _V) .: X ) by A4, A5, FUNCT_1:def 6;

hence y in G2 .edgesBetween ((F _V) .: X) by A7, GLIB_000:32; :: thesis: verum

end;assume y in rng ((F _E) | (G1 .edgesBetween X)) ; :: thesis: y in G2 .edgesBetween ((F _V) .: X)

then consider x being object such that

A3: ( x in dom ((F _E) | (G1 .edgesBetween X)) & ((F _E) | (G1 .edgesBetween X)) . x = y ) by FUNCT_1:def 3;

set v = (the_Source_of G1) . x;

set w = (the_Target_of G1) . x;

A4: ( x in the_Edges_of G1 & (the_Source_of G1) . x in X & (the_Target_of G1) . x in X ) by A2, A3, GLIB_000:31;

then ( (the_Source_of G1) . x in the_Vertices_of G1 & (the_Target_of G1) . x in the_Vertices_of G1 ) ;

then A5: ( (the_Source_of G1) . x in dom (F _V) & (the_Target_of G1) . x in dom (F _V) ) by A1, Def11;

A6: x in dom (F _E) by A1, A4, Def11;

x Joins (the_Source_of G1) . x,(the_Target_of G1) . x,G1 by A4, GLIB_000:def 13;

then (F _E) . x Joins (F _V) . ((the_Source_of G1) . x),(F _V) . ((the_Target_of G1) . x),G2 by A5, A6, Th4;

then A7: y Joins (F _V) . ((the_Source_of G1) . x),(F _V) . ((the_Target_of G1) . x),G2 by A3, FUNCT_1:47;

( (F _V) . ((the_Source_of G1) . x) in (F _V) .: X & (F _V) . ((the_Target_of G1) . x) in (F _V) .: X ) by A4, A5, FUNCT_1:def 6;

hence y in G2 .edgesBetween ((F _V) .: X) by A7, GLIB_000:32; :: thesis: verum

(F _E) | (G1 .edgesBetween X) is one-to-one by A1, FUNCT_1:52;

hence card (G1 .edgesBetween X) c= card (G2 .edgesBetween ((F _V) .: X)) by A2, A8, CARD_1:10; :: thesis: verum