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
  d in dom F /\ SD implies F/.d = (F*id SD)/.d
proof
  assume
A1: d in dom F /\ SD;
  then
A2: d in dom F by XBOOLE_0:def 4;
  (F qua Function).d = ((F*(id SD)) qua Function).d by A1,FUNCT_1:20;
  then
A3: F/.d = ((F*(id SD)) qua Function).d by A2,PARTFUN1:def 6;
A4: d in SD by A1,XBOOLE_0:def 4;
  then
A5: d in dom id SD by RELAT_1:45;
  (id SD)/.d in dom F by A2,A4,Th6;
  then d in dom (F*(id SD)) by A5,Th3;
  hence thesis by A3,PARTFUN1:def 6;
end;
