reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th7:
  for D be non empty set, F be PartFunc of D,REAL holds (0(#)F)"{0} = dom F
proof
  let D be non empty set, F be PartFunc of D,REAL;
  thus (0(#)F)"{0} c= dom F
  proof
    let x be object;
    assume
A1: x in (0(#)F)"{0};
    then reconsider d=x as Element of D;
    d in dom(0(#)F) by A1,FUNCT_1:def 7;
    hence thesis by VALUED_1:def 5;
  end;
  let x be object;
  assume
A2: x in dom F;
  then reconsider d=x as Element of D;
A3: d in dom(0(#)F) by A2,VALUED_1:def 5;
  then (0(#)F).d = 0*(F.d) by VALUED_1:def 5
    .= 0;
  then (0(#)F).d in {0} by TARSKI:def 1;
  hence thesis by A3,FUNCT_1:def 7;
end;
