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 Th69:
  2|^n mod 4 = 2 implies n = 1
  proof
    assume
A1: 2|^n mod 4 = 2;
    n = 0 or n >= 0+1 by NAT_1:13;
    then per cases by XXREAL_0:1;
    suppose n = 0;
      then 2|^n = 1 by NEWTON:4;
      hence thesis by A1,NAT_D:24;
    end;
    suppose n = 0+1;
      hence thesis;
    end;
    suppose n > 0+1;
      then n >= 1+1 by NAT_1:13;
      then 2|^2 divides 2|^n by NEWTON:89;
      hence thesis by A1,Lm3,INT_1:62;
    end;
  end;
