
theorem :: MSnsize:
for G being finite SimpleGraph, n being Nat
 holds size (MycielskianSeq G).n
     = 3|^n*(size G) + (3|^n - 2|^n)*(order G) + (n+1) block 3
proof
 let G be finite SimpleGraph, n being Nat;
 set g = order G; set s = size G;
  set MG = MycielskianSeq G;
  defpred P[Nat] means
   size ((MG)).$1 = 3|^$1*s + (3|^$1 - 2|^$1)*g + ($1+1) block 3;
A1: P[0] proof
     thus size ((MG).0)
        = 1*s + 0*g + 0 by Th113
       .= 3|^0*s + (1 - 1)*g + 0 by NEWTON:4
       .= 3|^0*s + (3|^0 - 1)*g + 0 by NEWTON:4
       .= 3|^0*s + (3|^0 - 2|^0)*g + 0 by NEWTON:4
       .= 3|^0*s + (3|^0 - 2|^0)*g + (0+1) block 3 by STIRL2_1:29;
    end;
A2: for n being Nat st P[n] holds P[n+1] proof
     let n be Nat such that
    A3: P[n];
    A4: n+1 >= 0+1 by XREAL_1:6;
    A5: 1/2*(2|^(n+1) - 2)
       = 1/2*(2*2|^n - 2*1) by NEWTON:6
      .= 2|^n -1;
     thus size ((MG).(n+1))
      = size (Mycielskian ((MG.n))) by Th115
     .= 3*(3|^n*s + (3|^n - 2|^n)*g + (n+1) block 3) + order (MG.n)
        by A3,Th92
     .= 3*3|^n*s + 3*(3|^n - 2|^n)*g + 3*((n+1) block 3) + order (MG.n)
     .= 3|^(n+1)*s + 3*(3|^n - 2|^n)*g + 3*((n+1) block 3) + order (MG.n)
        by NEWTON:6
     .= 3|^(n+1)*s + 3*(3|^n - 2|^n)*g + 3*((n+1) block 3) + (2|^n*g + 2|^n -1)
        by Th116
   .= 3|^(n+1)*s + 3*(3|^n - 2|^n)*g + 2|^n*g + (3*((n+1) block 3) + (2|^n -1))
   .= 3|^(n+1)*s + 3*(3|^n - 2|^n)*g + 2|^n*g + ((2+1)*((n+1) block (2+1))
      + ((n+1) block 2)) by A5,A4,STIRL2_1:47
   .= 3|^(n+1)*s + (3*3|^n*g - (2*2|^n*g+2|^n*g) + 2|^n*g) + ((n+1)+1) block 3
       by STIRL2_1:46
  .= 3|^(n+1)*s + (3*3|^n*g - (2|^(n+1)*g+2|^n*g) + 2|^n*g) + ((n+1)+1) block 3
       by NEWTON:6
  .= 3|^(n+1)*s + (3*3|^n*g - 2|^(n+1)*g - 2|^n*g + 2|^n*g) + ((n+1)+1) block 3
  .= 3|^(n+1)*s + (3|^(n+1)*g - 2|^(n+1)*g-2|^n*g+2|^n*g) + ((n+1)+1) block 3
     by NEWTON:6
     .= 3|^(n+1)*s + (3|^(n+1) - 2|^(n+1))*g + ((n+1)+1) block 3;
    end;
 for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
 hence size ((MG).n) = 3|^n*s + (3|^n - 2|^n)*g + (n+1) block 3;
end;
