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

theorem Th9:
  for D be non empty set, F be PartFunc of D,REAL holds abs(F)"{0} = F"{0}
proof
  let D be non empty set, F be PartFunc of D,REAL;
A1: dom abs(F) = dom F by VALUED_1:def 11;
  thus abs(F)"{0} c= F"{0}
  proof
    let x be object;
    assume
A2: x in abs(F)"{0};
    then reconsider r=x as Element of D;
    abs(F).r in {0} by A2,FUNCT_1:def 7;
    then |.F.r.| in {0} by VALUED_1:18;
    then |.F.r.| = 0 by TARSKI:def 1;
    then F.r = 0 by ABSVALUE:2;
    then
A3: F.r in {0} by TARSKI:def 1;
    r in dom abs(F) by A2,FUNCT_1:def 7;
    hence thesis by A1,A3,FUNCT_1:def 7;
  end;
  let x be object;
  assume
A4: x in F"{0};
  then reconsider r=x as Element of D;
  F.r in {0} by A4,FUNCT_1:def 7;
  then F.r = 0 by TARSKI:def 1;
  then |.F.r.| = 0 by ABSVALUE:2;
  then abs(F).r = 0 by VALUED_1:18;
  then
A5: abs(F).r in {0} by TARSKI:def 1;
  r in dom F by A4,FUNCT_1:def 7;
  hence thesis by A1,A5,FUNCT_1:def 7;
end;
