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 Th4:
  c in dom f & f/.c in dom s implies (s*f)/.c = s/.(f/.c)
proof
  assume c in dom f & f/.c in dom s;
  then c in dom (s*f) by Th3;
  hence thesis by Th3;
end;
