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 Th5:
  dom (-f) = dom f & for c st c in dom (-f) holds (-f)/.c = -f/.c
proof
  thus
A1: dom (-f) = dom f by VALUED_1:8;
  now
    let c;
    assume
A2: c in dom (-f);
    hence (-f)/.c = (-f).c by PARTFUN1:def 6
      .=-f.c by VALUED_1:8
      .=-f/.c by A1,A2,PARTFUN1:def 6;
  end;
  hence thesis;
end;
