let G be _finite edgeless _Graph; :: thesis: for H being Subgraph of G ex p being non empty Graph-yielding _finite edgeless 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: ex p being non empty Graph-yielding _finite edgeless 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) ) ) )
G .size() = 0 by Th49
.= H .size() by Th49 ;
then consider p being non empty Graph-yielding _finite FinSequence such that
A1: ( p . 1 == H & p . (len p) = G & len p = ((G .order()) - (H .order())) + 1 ) and
A2: 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 Th60;
defpred S₁[ Nat] means for n being Element of dom p st $1 = n holds
p . n is edgeless ;
A3: S₁[1] by A1, Th52;
A4: 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 A5: S₁[k] ; :: thesis: S₁[k + 1]
let m be Element of dom p; :: thesis: ( k + 1 = m implies p . m is edgeless )
assume A6: k + 1 = m ; :: thesis: p . m is edgeless
then A7: k + 1 <= len p by FINSEQ_3:25;
then A8: (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 A8, FINSEQ_3:25;
(k + 1) - 1 <= (len p) - 1 by A7, XREAL_1:9;
then consider v being Vertex of G such that
A9: ( p . (n + 1) is addVertex of p . n,v & not v in the_Vertices_of (p . n) ) by A2;
p . n is edgeless by A5;
hence p . m is edgeless by A6, A9; :: thesis: verum

end;

A10: for k being non zero Nat holds S₁[k] from NAT_1:sch 10(A3, A4);
for x being Element of dom p holds p . x is edgeless

proof

let x be Element of dom p; :: thesis: p . x is edgeless
x is non zero Nat by FINSEQ_3:25;
hence p . x is edgeless by A10; :: thesis: verum

end;

then p is edgeless ;
then reconsider p = p as non empty Graph-yielding _finite edgeless FinSequence ;
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 & 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) ) ) ) by A1, A2; :: thesis: verum