let G be _finite _Graph; :: thesis: for H being Subgraph of G st G .size() = H .size() holds
ex p being non empty Graph-yielding _finite FinSequence st
( p . 1 == H & p . (len p) = G & len p = ((G .order()) - (H .order())) + 1 & ( for n being Element of dom p st n <= (len p) - 1 holds
ex v being Vertex of G st
( p . (n + 1) is addVertex of p . n,v & not v in the_Vertices_of (p . n) ) ) )
let H be Subgraph of G; :: thesis: ( G .size() = H .size() implies ex p being non empty Graph-yielding _finite FinSequence st
( p . 1 == H & p . (len p) = G & len p = ((G .order()) - (H .order())) + 1 & ( for n being Element of dom p st n <= (len p) - 1 holds
ex v being Vertex of G st
( p . (n + 1) is addVertex of p . n,v & not v in the_Vertices_of (p . n) ) ) ) )
assume A1: G .size() = H .size() ; :: thesis: ex p being non empty Graph-yielding _finite FinSequence st
( p . 1 == H & p . (len p) = G & len p = ((G .order()) - (H .order())) + 1 & ( for n being Element of dom p st n <= (len p) - 1 holds
ex v being Vertex of G st
( p . (n + 1) is addVertex of p . n,v & not v in the_Vertices_of (p . n) ) ) )
set V = (the_Vertices_of G) \ (the_Vertices_of H);
G is addVertices of H,(the_Vertices_of G) \ (the_Vertices_of H) by A1, Th34;
then consider p being non empty Graph-yielding FinSequence such that
A2: ( p . 1 == H & p . (len p) = G & len p = (card (((the_Vertices_of G) \ (the_Vertices_of H)) \ (the_Vertices_of H))) + 1 ) and
A3: for n being Element of dom p st n <= (len p) - 1 holds
ex v being Vertex of G st
( p . (n + 1) is addVertex of p . n,v & not v in the_Vertices_of (p . n) ) by Th59;
defpred S₁[ Nat] means for n being Element of dom p st $1 = n holds
p . n is _finite ;
A4: S₁[1] by A2, GLIB_000:89;
A5: for k being non zero Nat st S₁[k] holds
S₁[k + 1]

proof

let k be non zero Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A6: S₁[k] ; :: thesis: S₁[k + 1]
let m be Element of dom p; :: thesis: ( k + 1 = m implies p . m is _finite )
assume A7: k + 1 = m ; :: thesis: p . m is _finite
then A8: k + 1 <= len p by FINSEQ_3:25;
then A9: (k + 1) - 1 <= (len p) - 0 by XREAL_1:13;
1 <= k by NAT_1:14;
then reconsider n = k as Element of dom p by A9, FINSEQ_3:25;
(k + 1) - 1 <= (len p) - 1 by A8, XREAL_1:9;
then consider v being Vertex of G such that
A10: ( p . (n + 1) is addVertex of p . n,v & not v in the_Vertices_of (p . n) ) by A3;
p . n is _finite by A6;
hence p . m is _finite by A7, A10; :: thesis: verum

end;

A11: for k being non zero Nat holds S₁[k] from NAT_1:sch 10(A4, A5);
for x being Element of dom p holds p . x is _finite

proof

let x be Element of dom p; :: thesis: p . x is _finite
x is non zero Nat by FINSEQ_3:25;
hence p . x is _finite by A11; :: thesis: verum

end;

then reconsider p = p as non empty Graph-yielding _finite FinSequence by GLIB_000:def 66;
take p ; :: thesis: ( p . 1 == H & p . (len p) = G & len p = ((G .order()) - (H .order())) + 1 & ( for n being Element of dom p st n <= (len p) - 1 holds
ex v being Vertex of G st
( p . (n + 1) is addVertex of p . n,v & not v in the_Vertices_of (p . n) ) ) )
thus ( p . 1 == H & p . (len p) = G ) by A2; :: thesis: ( len p = ((G .order()) - (H .order())) + 1 & ( for n being Element of dom p st n <= (len p) - 1 holds
ex v being Vertex of G st
( p . (n + 1) is addVertex of p . n,v & not v in the_Vertices_of (p . n) ) ) )
((the_Vertices_of G) \ (the_Vertices_of H)) \ (the_Vertices_of H) = (the_Vertices_of G) \ ((the_Vertices_of H) \/ (the_Vertices_of H)) by XBOOLE_1:41
.= (the_Vertices_of G) \ (the_Vertices_of H) ;
hence len p = ((card (the_Vertices_of G)) - (card (the_Vertices_of H))) + 1 by A2, CARD_2:44
.= ((G .order()) - (card (the_Vertices_of H))) + 1 by GLIB_000:def 24
.= ((G .order()) - (H .order())) + 1 by GLIB_000:def 24 ;
:: thesis: for n being Element of dom p st n <= (len p) - 1 holds
ex v being Vertex of G st
( p . (n + 1) is addVertex of p . n,v & not v in the_Vertices_of (p . n) )
thus for n being Element of dom p st n <= (len p) - 1 holds
ex v being Vertex of G st
( p . (n + 1) is addVertex of p . n,v & not v in the_Vertices_of (p . n) ) by A3; :: thesis: verum