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

theorem Th10:
  for D be non empty set, F be PartFunc of D,REAL, r st r<
  0 holds abs(F)"{r} = {}
proof
  let D be non empty set, F be PartFunc of D,REAL, r;
  assume
A1: r<0;
  set x = the Element of abs(F)"{r};
  assume
A2: abs(F)"{r} <> {};
  then reconsider x as Element of D by TARSKI:def 3;
  abs(F).x in {r} by A2,FUNCT_1:def 7;
  then |.F.x.| in {r} by VALUED_1:18;
  then |.F.x.| = r by TARSKI:def 1;
  hence contradiction by A1,COMPLEX1:46;
end;
