
theorem Th2:
  for f being Function, x, y being object holds
  support (f +* (x.-->y)) c= support f \/ {x}
proof
  let f be Function, x, y be object;
  let a be object;
  assume a in support (f +* (x.-->y));
  then
A1: (f +* (x.-->y)).a <> 0 by PRE_POLY:def 7;
  per cases;
  suppose
    a = x;
    then a in {x} by TARSKI:def 1;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    a <> x;
    then (f +* (x.-->y)).a = f.a by FUNCT_4:83;
    then a in support f by A1,PRE_POLY:def 7;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
