theorem
  a,(a+b)! are_coprime implies a = 1 or (a = 0 & (b = 0 or b = 1))
  proof
    assume
    A1: a,(a+b)! are_coprime;
    per cases;
    suppose
      B1: a = 0; then
      b <= 1 by A1,Th23; then
      b < 1+1 by NAT_1:13;
      hence thesis by B1,NAT_1:23;
    end;
    suppose a>0;
      hence thesis by A1,Lm3,NEWTON:49;
    end;
  end;
