
theorem Th116:
for G being finite SimpleGraph, n being Nat
 holds order ((MycielskianSeq G).n) = 2|^n*(order G) + 2|^n - 1
proof
 let G be finite SimpleGraph, n being Nat;
 set g = order G;
  set MG = MycielskianSeq G;
  defpred P[Nat] means
   order ((MG).$1) = 2|^$1*g + 2|^$1 - 1;
A1: P[0] proof
     thus order (MG.0)
        = g + 1 - 1 by Th113
       .= 1*g+2|^0-1 by NEWTON:4
       .= 2|^0*g+2|^0-1 by NEWTON:4;
    end;
A2: for n being Nat st P[n] holds P[n+1] proof
     let n be Nat such that
    A3: P[n];
     thus order ((MG).(n+1))
      = order (Mycielskian ((MG.n))) by Th115
     .= 2*(2|^n*g + 2|^n - 1) + 1 by A3,Th89
     .= 2*(2|^n)*g + 2*(2|^n) - 2*1 + 1
     .= 2|^(n+1)*g + 2*(2|^n) - 2*1 + 1 by NEWTON:6
     .= 2|^(n+1)*g + 2|^(n+1) - 2 + 1 by NEWTON:6
     .= 2|^(n+1)*g + 2|^(n+1) - 1;
    end;
  for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
 hence order ((MG).n) = 2|^n*g + 2|^n - 1;
end;
