reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th10:
  |.f.|"{0} = f"{0} & (-f)"{0} = f"{0}
proof
A1: dom |.f.| = dom f by VALUED_1:def 11;
  now
    let c;
    thus c in (|.f.|)"{0} implies c in f"{0c}
    proof
      assume
A2:   c in (|.f.|)"{0};
      then c in dom (|.f.|) by FUNCT_1:def 7;
      then
A3:   c in dom f by VALUED_1:def 11;
      (|.f.|).c in {0} by A2,FUNCT_1:def 7;
      then (|.f.|).c = 0 by TARSKI:def 1;
      then
A4:   |.(f.c).| = 0 by VALUED_1:18;
      c in dom (|.f.|) by A2,FUNCT_1:def 7;
      then f.c = f/.c by A1,PARTFUN1:def 6;
      then (f/.c) = 0c by A4,COMPLEX1:45;
      then (f/.c) in {0c} by TARSKI:def 1;
      hence thesis by A3,PARTFUN2:26;
    end;
    assume
A5: c in (f)"{0c};
    then (f/.c) in {0c} by PARTFUN2:26;
    then
A6: |.(f/.c).| = 0 by COMPLEX1:44,TARSKI:def 1;
A7: c in dom f by A5,PARTFUN2:26;
    then f.c = f/.c by PARTFUN1:def 6;
    then (|.f.|).c = 0 by A6,VALUED_1:18;
    then
A8: (|.f.|).c in {0} by TARSKI:def 1;
    c in dom (|.f.|) by A7,VALUED_1:def 11;
    hence c in (|.f.|)"{0} by A8,FUNCT_1:def 7;
  end;
  hence (|.f.|)"{0} = f"{0} by SUBSET_1:3;
  now
    let c;
    thus c in (-f)"{0c} implies c in f"{0c}
    proof
      assume
A9:   c in (-f)"{0c};
      then
A10:  c in dom (-f) by PARTFUN2:26;
      (-f)/.c in {0c} by A9,PARTFUN2:26;
      then (-f)/.c = 0c by TARSKI:def 1;
      then --((f/.c)) = -0c by A10,Th5;
      then
A11:  (f/.c) in {0c} by TARSKI:def 1;
      c in dom f by A10,Th5;
      hence thesis by A11,PARTFUN2:26;
    end;
    assume
A12: c in (f)"{0c};
    then (f/.c) in {0c} by PARTFUN2:26;
    then
A13: (f/.c) = 0c by TARSKI:def 1;
A14: c in dom f by A12,PARTFUN2:26;
    then c in dom (-f) by Th5;
    then (-f)/.c = -0c by A13,Th5;
    then
A15: (-f)/.c in {0c} by TARSKI:def 1;
    c in dom (-f) by A14,Th5;
    hence c in (-f)"{0c} by A15,PARTFUN2:26;
  end;
  hence thesis by SUBSET_1:3;
end;
