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
  for f ex g being Function of C,D st for c st c in dom f holds g.c = f /.c
proof
  let f;
  consider g being Function of C,D such that
A1: for x being object st x in dom f holds g.x = (f qua Function).x
    by FUNCT_2:71;
  take g;
  let c;
  assume
A2: c in dom f;
  then g.c = (f qua Function).c by A1;
  hence thesis by A2,PARTFUN1:def 6;
end;
