reserve x,y,z for Real,
  a,b,c,d,e,f,g,h for Nat,
  k,l,m,n,m1,n1,m2,n2 for Integer,
  q for Rational;
reserve fs,fs1,fs2,fs3 for FinSequence;
reserve D for non empty set,
  v,v1,v2,v3 for object,
  fp for FinSequence of NAT,
  fr,fr1,fr2 for FinSequence of INT,
  ft for FinSequence of REAL;
reserve x,y,t for Integer;
reserve n for Nat;

theorem
  k mod n = 0 implies - (k div n) = (-k) div n
proof
  assume
A1: k mod n = 0;
  per cases;
  suppose
A2: n > 0;
    then n divides k by A1,INT_1:62;
    then
A3: k/n is Integer by A2,Th17;
    thus - (k div n) = - [\ k / n /] by INT_1:def 9
    .= [\ (-k) / n /] by A3,INT_1:64
    .= (-k) div n by INT_1:def 9;
  end;
  suppose n = 0;
    hence thesis;
  end;
end;
