reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem
  b>0 & a,b! are_coprime implies a,b are_coprime
  proof
    assume
    A1: b > 0 & a,b! are_coprime;
    per cases;
    suppose
      a = 0; then
      b <= 1 by A1,Th23; then
      b < 1+1 by NAT_1:13; then
      b = 1 or b = 0 by NAT_1:23;
      hence thesis by A1,NEWTON:51;
    end;
    suppose a>0;
      hence thesis by A1,Lm4;
    end;
  end;
