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

theorem Th8:
  for D be non empty set, F be PartFunc of D,REAL, r st 0<
  r holds abs(F)"{r} = F"{-r,r}
proof
  let D be non empty set, F be PartFunc of D,REAL, r;
  assume
A1: 0<r;
A2: dom abs(F) = dom F by VALUED_1:def 11;
  thus abs(F)"{r} c= F"{-r,r}
  proof
    let x be object;
    assume
A3: x in abs(F)"{r};
    then reconsider rr = x as Element of D;
    abs(F).rr in {r} by A3,FUNCT_1:def 7;
    then |.F.rr.| in {r} by VALUED_1:18;
    then
A4: |.F.rr.| = r by TARSKI:def 1;
A5: rr in dom abs(F) by A3,FUNCT_1:def 7;
    now
      per cases;
      case
        0<=F.rr;
        then F.rr = r by A4,ABSVALUE:def 1;
        then F.rr in {-r,r} by TARSKI:def 2;
        hence thesis by A2,A5,FUNCT_1:def 7;
      end;
      case
        F.rr<0;
        then -F.rr = r by A4,ABSVALUE:def 1;
        then F.rr in {-r,r} by TARSKI:def 2;
        hence thesis by A2,A5,FUNCT_1:def 7;
      end;
    end;
    hence thesis;
  end;
  let x be object;
  assume
A6: x in F"{-r,r};
  then reconsider rr = x as Element of D;
A7: rr in dom F by A6,FUNCT_1:def 7;
A8: F.rr in {-r,r} by A6,FUNCT_1:def 7;
  now
    per cases by A8,TARSKI:def 2;
    case
      F.rr = -r;
      then r = |.-F.rr.| by A1,ABSVALUE:def 1
        .= |.F.rr.| by COMPLEX1:52
        .= abs(F).rr by VALUED_1:18;
      then abs(F).rr in {r} by TARSKI:def 1;
      hence thesis by A2,A7,FUNCT_1:def 7;
    end;
    case
      F.rr = r;
      then r = |.F.rr.| by A1,ABSVALUE:def 1
        .= abs(F).rr by VALUED_1:18;
      then abs(F).rr in {r} by TARSKI:def 1;
      hence thesis by A2,A7,FUNCT_1:def 7;
    end;
  end;
  hence thesis;
end;
