reserve x,y,X,Y for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,f1,f2,f3,g,g1 for PartFunc of C,COMPLEX;
reserve r1,r2,p1 for Real;
reserve r,q,cr1,cr2 for Complex;

theorem Th11:
  dom (f^^) = dom (f|(dom (f^)))
proof
A1: dom (f^) = dom f \ f"{0c} by Def2;
  thus dom (f^^) = dom (f^) \(f^)"{0c} by Def2
    .= dom (f^) \ {} by Th9
    .= dom f /\ dom (f^) by A1,XBOOLE_1:28,36
    .= dom (f|(dom (f^))) by RELAT_1:61;
end;
