reserve
  S for (4,1) integer bool-correct non empty non void BoolSignature,
  X for non-empty ManySortedSet of the carrier of S,
  T for vf-free integer all_vars_including inheriting_operations free_in_itself
  (X,S)-terms VarMSAlgebra over S,
  C for (4,1) integer bool-correct non-empty image of T,
  G for basic GeneratorSystem over S,X,T,
  A for IfWhileAlgebra of the generators of G,
  I for integer SortSymbol of S,
  x,y,z,m for pure (Element of (the generators of G).I),
  b for pure (Element of (the generators of G).the bool-sort of S),
  t,t1,t2 for Element of T,I,
  P for Algorithm of A,
  s,s1,s2 for Element of C-States(the generators of G);
reserve
  f for ExecutionFunction of A, C-States(the generators of G),
  (\falseC)-States(the generators of G, b);
reserve u for ManySortedFunction of FreeGen T, the Sorts of C;

theorem Th64:
  for i,j being Integer, a,b being Element of C,I st a = i & b = j & j <> 0
  holds a mod b = i mod j
  proof
    let i,j be Integer;
    let a,b be Element of C,I;
    assume A1: a = i;
    assume A2: b = j;
    assume A3: j <> 0;
    then a div b = i div j by A1,A2,AOFA_A00:55;
    then (a div b)*b = (i div j)*j by A2,AOFA_A00:55;
    then a-(a div b)*b = i-(i div j)*j by A1,Th63;
    hence a mod b = i mod j by A3,INT_1:def 10;
  end;
