let G be finite _Graph; :: thesis: for S being VNumberingSeq of G
for n being Nat holds S . n is one-to-one
let S be VNumberingSeq of G; :: thesis: for n being Nat holds S . n is one-to-one
let n be Nat; :: thesis: S . n is one-to-one
defpred S₁[ Nat] means S . $1 is one-to-one ;
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

set GN = S .Lifespan() ;
let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
set w = S .PickedAt k;
set CK1 = S . (k + 1);
set CSK = S . k;
set VLK = S . k;
set VL1 = S . (k + 1);

per cases ( k < S .Lifespan() or k >= S .Lifespan() ) ;

suppose A3: k < S .Lifespan() ; :: thesis: S₁[k + 1]

set wf = (S .PickedAt k) .--> ((S .Lifespan() ) -' k);

A4: now

assume A5: (S .Lifespan() ) -' k in rng (S . k) ; :: thesis: contradiction
rng (S . k) = (Seg (S .Lifespan() )) \ (Seg ((S .Lifespan() ) -' k)) by Th14;
hence contradiction by A5, Th3; :: thesis: verum

end;

rng ((S .PickedAt k) .--> ((S .Lifespan() ) -' k)) = {((S .Lifespan() ) -' k)} by FUNCOP_1:14;
then A6: rng ((S .PickedAt k) .--> ((S .Lifespan() ) -' k)) misses rng (S . k) by A4, ZFMISC_1:56;
S . (k + 1) = (S . k) +* ((S .PickedAt k) .--> ((S .Lifespan() ) -' k)) by A3, Def10;
hence S₁[k + 1] by A2, A6, FUNCT_4:98; :: thesis: verum

end;

suppose A7: k >= S .Lifespan() ; :: thesis: S₁[k + 1]

k <= k + 1 by NAT_1:13;
hence S₁[k + 1] by A2, A7, Th10; :: thesis: verum

end;

end;

end;

A8: S₁[ 0 ] by Def9;
for k being Nat holds S₁[k] from NAT_1:sch 2(A8, A1);
hence S . n is one-to-one ; :: thesis: verum