reserve a, b, n for Nat,
  r for Real,
  f for FinSequence of REAL;
reserve p for Prime;

theorem Th16:
  for X being set, b being real-valued ManySortedSet of X st a <>
  0 holds support b = support (a*b)
proof
  let X be set, b be real-valued ManySortedSet of X such that
A1: a <> 0;
  hereby
    let x be object;
    assume x in support b;
    then b.x <> 0 by PRE_POLY:def 7;
    then a*b.x <> 0 by A1;
    then (a*b).x <> 0 by Def2;
    hence x in support (a*b) by PRE_POLY:def 7;
  end;
  let x be object;
  assume x in support (a*b);
  then (a*b).x <> 0 by PRE_POLY:def 7;
  then a*b.x <> 0 by Def2;
  then b.x <> 0;
  hence thesis by PRE_POLY:def 7;
end;
