reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem
  not 6 divides 2|^6-2 & 6 divides 3|^6-3 &
  not ex n being Nat st n < 6 & not n divides 2|^n-2 & n divides 3|^n-3
  proof
    2|^6-2 = 2*2*2*2*2*2-2 by Th2
    .= 6*10+2;
    hence not 6 divides 2|^6-2 by Th34;
    3|^6-3 = 3*3*3*3*3*3-3 by Th2
    .= 6*121;
    hence 6 divides 3|^6-3;
    let n be Nat such that
A1: n < 6 and
A2: not n divides 2|^n-2;
    n = 0 or ... or n = 6-1 by A1,Th7;
    then
A3: n = 0 or ... or n = 5;
    per cases by A2,A3,INT_2:28,PEPIN:41,59,INT_2:12,NEWTON02:138;
    suppose n = 0;
      hence not n divides 3|^n-3 by NEWTON:4;
    end;
    suppose
A4:   n = 4;
      3|^4-3 = 3*3*3*3-3 by POLYEQ_5:3
      .= 4*19+2;
      hence not n divides 3|^n-3 by A4,Th32;
    end;
  end;
