reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th2:
  for f being finite-support Function holds rng f c= rng(f|support f) \/ {0}
  proof
    let f be finite-support Function;
    set g = f|support f;
    let y be object;
    assume y in rng f;
    then consider x being object such that
A1: x in dom f and
A2: f.x = y by FUNCT_1:def 3;
    per cases;
    suppose f.x = 0;
      then f.x in {0} by TARSKI:def 1;
      hence thesis by A2,XBOOLE_0:def 3;
    end;
    suppose f.x <> 0;
      then
A3:   x in support f by PRE_POLY:def 7;
      then
A4:   f.x = g.x by FUNCT_1:49;
      x in dom g by A1,A3,RELAT_1:57;
      then g.x in rng g by FUNCT_1:def 3;
      hence thesis by A2,A4,XBOOLE_0:def 3;
    end;
  end;
