reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th71:
  n is odd & 3|^n - 2|^m = 1 implies n = m = 1
  proof
    assume that
A1: n is odd and
A2: 3|^n - 2|^m = 1;
A3: 3|^n = 1 + 2|^m by A2;
    consider k such that
A4: n = 2*k+1 by A1,ABIAN:9;
    3|^(2*k),1 are_congruent_mod 4 by Th68;
    then 3|^(2*k)*3,1*3 are_congruent_mod 4 by Lm13,INT_1:18;
    then 3|^n,3 are_congruent_mod 4 by A4,NEWTON:6;
    then 2|^m,3-1 are_congruent_mod 4 by A2;
    then 2|^m mod 4 = 2 mod 4 by NAT_D:64
    .= 2 by NAT_D:24;
    then
A5: m = 1 by Th69;
    then 3|^n = 3|^1 by A3;
    hence thesis by A5,PEPIN:30;
  end;
