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;
