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
  c in dom f & c in X implies f/.c in rng (f|X)
proof
  assume that
A1: c in dom f and
A2: c in X;
  (f qua Function).c in rng (f|X) by A1,A2,FUNCT_1:50;
  hence thesis by A1,PARTFUN1:def 6;
end;
