reserve a,b,p,k,l,m,n,s,h,i,j,t,i1,i2 for natural Number;

theorem
  for i,j being Integer st i>=0 & j>0 holds i gcd j= j gcd (i mod j)
proof
  let i,j be Integer;
  assume that
A1: i>=0 and
A2: j>0;
A3: |.j.|> 0 by A2,ABSVALUE:def 1;
  thus i gcd j = |.i.| gcd |.j.| by INT_2:34
    .=|.j.| gcd (|.i.| mod |.j.|) by A3,Th28
    .=|.j.| gcd |.|.i.| mod |.j.|.| by ABSVALUE:def 1
    .=j gcd (|.i.| mod |.j.|) by INT_2:34
    .=j gcd (i mod j) by A1,A2,INT_2:32;
end;
