reserve x for object, X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for complex-valued Function;
reserve r,p for Complex;

theorem Th5:
  (abs(f))"{0} = f"{0} & (-f)"{0} = f"{0}
proof
  now
    let c be object;
    reconsider cc = c as object;
    thus c in (abs(f))"{0} implies c in f"{0}
    proof
      assume
A1:   c in (abs(f))"{0};
      then (abs(f)).c in {0} by FUNCT_1:def 7;
      then (abs(f)).c = 0 by TARSKI:def 1;
      then |.f.cc.| = 0 by VALUED_1:18;
      then f.c = 0 by COMPLEX1:45;
      then
A2:   f.c in {0} by TARSKI:def 1;
      c in dom (abs(f)) by A1,FUNCT_1:def 7;
      then c in dom f by VALUED_1:def 11;
      hence thesis by A2,FUNCT_1:def 7;
    end;
    assume
A3: c in (f)"{0};
    then f.c in {0} by FUNCT_1:def 7;
    then f.c = 0 by TARSKI:def 1;
    then |.f.cc.| = 0 by ABSVALUE:2;
    then (abs(f)).c = 0 by VALUED_1:18;
    then
A4: (abs(f)).c in {0} by TARSKI:def 1;
    c in dom f by A3,FUNCT_1:def 7;
    then c in dom (abs(f)) by VALUED_1:def 11;
    hence c in (abs(f))"{0} by A4,FUNCT_1:def 7;
  end;
  hence (abs(f))"{0} = f"{0} by TARSKI:2;
  now
    let c be object;
    reconsider cc = c as object;
    thus c in (-f)"{0} implies c in f"{0}
    proof
      assume
A5:   c in (-f)"{0};
      then (-f).c in {0} by FUNCT_1:def 7;
      then (-f).c = 0 by TARSKI:def 1;
      then --(f.cc) = -In(0,REAL) by VALUED_1:8;
      then
A6:   f.c in {0} by TARSKI:def 1;
      c in dom (-f) by A5,FUNCT_1:def 7;
      then c in dom f by VALUED_1:8;
      hence thesis by A6,FUNCT_1:def 7;
    end;
    assume
A7: c in (f)"{0};
    then f.c in {0} by FUNCT_1:def 7;
    then f.c = 0 by TARSKI:def 1;
    then (-f).c = -In(0,REAL) by VALUED_1:8;
    then
A8: (-f).c in {0} by TARSKI:def 1;
    c in dom f by A7,FUNCT_1:def 7;
    then c in dom (-f) by VALUED_1:8;
    hence c in (-f)"{0} by A8,FUNCT_1:def 7;
  end;
  hence thesis by TARSKI:2;
end;
