let G2 be _Graph; for v1, v2 being Vertex of G2
for e being object
for G1 being addEdge of G2,v1,e,v2 st not e in the_Edges_of G2 holds
( G1 .order() = G2 .order() & G1 .size() = (G2 .size()) +` 1 )
let v1, v2 be Vertex of G2; for e being object
for G1 being addEdge of G2,v1,e,v2 st not e in the_Edges_of G2 holds
( G1 .order() = G2 .order() & G1 .size() = (G2 .size()) +` 1 )
let e be object ; for G1 being addEdge of G2,v1,e,v2 st not e in the_Edges_of G2 holds
( G1 .order() = G2 .order() & G1 .size() = (G2 .size()) +` 1 )
let G1 be addEdge of G2,v1,e,v2; ( not e in the_Edges_of G2 implies ( G1 .order() = G2 .order() & G1 .size() = (G2 .size()) +` 1 ) )
assume A1:
not e in the_Edges_of G2
; ( G1 .order() = G2 .order() & G1 .size() = (G2 .size()) +` 1 )
then A2:
( the_Vertices_of G1 = the_Vertices_of G2 & the_Edges_of G1 = (the_Edges_of G2) \/ {e} )
by Def11;
thus G1 .order() =
card (the_Vertices_of G1)
by GLIB_000:def 24
.=
G2 .order()
by A2, GLIB_000:def 24
; G1 .size() = (G2 .size()) +` 1
thus G1 .size() =
card (the_Edges_of G1)
by GLIB_000:def 25
.=
(card (the_Edges_of G2)) +` (card {e})
by A1, A2, CARD_2:35, ZFMISC_1:50
.=
(G2 .size()) +` (card {e})
by GLIB_000:def 25
.=
(G2 .size()) +` 1
by CARD_2:42
; verum