set MG = MycielskianSeq G;
defpred S₂[ Nat] means (MycielskianSeq G) . G is with_finite_clique# ;
consider myc being Function such that
A1: MycielskianSeq G = myc and
A2: myc . 0 = G and
A3: for k being Nat
for G being SimpleGraph st G = myc . k holds
myc . (k + 1) = Mycielskian G by Def26;
A4: S₂[ 0 ] by A1, A2;
A5: for k being Nat st S₂[k] holds
S₂[k + 1]

proof

let k be Nat; :: thesis: ( S₂[k] implies S₂[k + 1] )
assume S₂[k] ; :: thesis: S₂[k + 1]
then reconsider H = (MycielskianSeq G) . k as with_finite_clique# SimpleGraph ;
(MycielskianSeq G) . (k + 1) = Mycielskian H by A1, A3;
hence S₂[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₂[k] from NAT_1:sch 2(A4, A5);
hence (MycielskianSeq G) . n is with_finite_clique# ; :: thesis: verum