reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th31:
  for a,b being odd Nat st a-b = 2 holds a,b are_coprime
  proof
    let a,b be odd Nat such that
A1: a-b = 2;
    assume not a,b are_coprime;
    then consider k being Nat such that
A2: k divides a & k divides b and
A3: k <> 1 by PYTHTRIP:def 1;
    k <= 2 by A1,A2,INT_5:1,INT_2:27;
    then k = 0 or ... or k = 2;
    hence contradiction by A2,A3;
  end;
