reserve a,b,c,k,l,m,n for Nat,
  i,j,x,y for Integer;

theorem Th15:
  x <> 0 & i >= 0 implies i*x gcd i*y = i*(x gcd y)
proof
  assume that
A1: x <> 0 and
A2: i >= 0;
  consider a2,b2 being Integer such that
A3: x = (x gcd y)*a2 & y = (x gcd y)*b2 and
A4: a2,b2 are_coprime by A1,INT_2:23;
  i*x = (i*(x gcd y))*a2 & i*y = (i*(x gcd y))*b2 by A3;
  then
A5: i*x gcd i*y = |.i*(x gcd y).| by A4,INT_2:24
    .= |.i.|*|.(x gcd y).| by COMPLEX1:65;
  i = |.i.| by A2,ABSVALUE:def 1;
  hence thesis by A5,ABSVALUE:def 1;
end;
