let G be SimpleGraph; :: thesis: for n being Nat holds (MycielskianSeq G) . n is SimpleGraph
let n be Nat; :: thesis: (MycielskianSeq G) . n is SimpleGraph
set MG = MycielskianSeq G;
defpred S₁[ Nat] means (MycielskianSeq G) . $1 is SimpleGraph;
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: S₁[ 0 ] by A1, A2;
A5: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume S₁[k] ; :: thesis: S₁[k + 1]
then reconsider H = (MycielskianSeq G) . k as SimpleGraph ;
(MycielskianSeq G) . (k + 1) = Mycielskian H by A1, A3;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A5);
hence (MycielskianSeq G) . n is SimpleGraph ; :: thesis: verum