reserve r1,r2,r3 for non negative Real;
reserve n,m1 for Nat;
reserve s for Real;
reserve cn,cd,i1,j1 for Integer;
reserve r for irrational Real;
reserve q for Rational;

theorem Th13:
  for f being Function,
      o1,o2 being object holds
  o1 in ZeroPointSet f iff o1 in dom f & f.o1 = 0
  proof
   let f be Function,
      o1,o2 be object;
   hereby
     assume
A1:  o1 in ZeroPointSet f;
     then not o1 in support f by XBOOLE_0:def 5;
     hence o1 in dom f & f.o1 = 0 by PRE_POLY:def 7,A1;
   end;
     assume that
A1:  o1 in dom f and
A2:  f.o1 = 0;
     not o1 in support f by A2,PRE_POLY:def 7;
     hence thesis by A1,XBOOLE_0:def 5;
   end;
