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;
