defpred S₂[ Nat] means ( clique# (Mycielskian $1) = 2 & chromatic# (Mycielskian $1) = $1 + 2 );
A1: clique# (Mycielskian 0) = clique# (CompleteRelStr 2) by Th49
.= 2 by Th33 ;
chromatic# (Mycielskian 0) = chromatic# (CompleteRelStr 2) by Th49
.= 2 by Th35 ;
then A2: S₂[ 0 ] by A1;
A3: for n being Nat st S₂[n] holds
S₂[n + 1]

proof

let n be Nat; :: thesis: ( S₂[n] implies S₂[n + 1] )
assume that
A4: clique# (Mycielskian n) = 2 and
A5: chromatic# (Mycielskian n) = n + 2 ; :: thesis: S₂[n + 1]
A6: Mycielskian (n + 1) = Mycielskian (Mycielskian n) by Th49;
thus clique# (Mycielskian (n + 1)) = 2 by A4, A6, Th46; :: thesis: chromatic# (Mycielskian (n + 1)) = (n + 1) + 2
thus chromatic# (Mycielskian (n + 1)) = 1 + (chromatic# (Mycielskian n)) by A6, Th48
.= (n + 1) + 2 by A5 ; :: thesis: verum

end;

for n being Nat holds S₂[n] from NAT_1:sch 2(A2, A3);
hence for n being Nat holds
( clique# (Mycielskian n) = 2 & chromatic# (Mycielskian n) = n + 2 ) ; :: thesis: verum