reserve a,b,c for Integer;
reserve i,j,k,l for Nat;
reserve n for Nat;
reserve a,b,c,d,a1,b1,a2,b2,k,l for Integer;
reserve p,p1,q,l for Nat;

theorem Th27:
  0 < b & a divides b implies a <= b
proof
  assume
A1: 0 < b;
  assume a divides b;
  then consider c such that
A2: b = a * c;
  per cases;
  suppose
    a <= 0;
    hence thesis by A1;
  end;
  suppose
A3: a > 0;
    then c > 0 by A1,A2;
    then c >= 0+1 by INT_1:7;
    hence thesis by A2,A3,XREAL_1:151;
  end;
end;
