reserve x,y,X,Y for set;
reserve C,D,E for non empty set;
reserve SC for Subset of C;
reserve SD for Subset of D;
reserve SE for Subset of E;
reserve c,c1,c2 for Element of C;
reserve d,d1,d2 for Element of D;
reserve e for Element of E;
reserve f,f1,g for PartFunc of C,D;
reserve t for PartFunc of D,C;
reserve s for PartFunc of D,E;
reserve h for PartFunc of C,E;
reserve F for PartFunc of D,D;

theorem Th2:
  y in rng f iff ex c st c in dom f & y = f/.c
proof
  thus y in rng f implies ex c st c in dom f & y = f/.c
  proof
    assume y in rng f;
    then consider x being object such that
A1: x in dom f and
A2: y = (f qua Function).x by FUNCT_1:def 3;
    reconsider x as Element of C by A1;
    take x;
    thus thesis by A1,A2,PARTFUN1:def 6;
  end;
  given c such that
A3: c in dom f and
A4: y = f/.c;
  (f qua Function).c in rng f by A3,FUNCT_1:def 3;
  hence thesis by A3,A4,PARTFUN1:def 6;
end;
