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 Th33:
  f^^ = f|(dom (f^))
proof
A1: dom (f^^) = dom (f|(dom (f^))) by Th11;
  now
    let c;
    assume
A2: c in dom (f^^);
    then c in dom f /\ dom (f^) by A1,RELAT_1:61;
    then
A3: c in dom (f^) by XBOOLE_0:def 4;
    thus (f^^)/.c = ((f^)/.c)" by A2,Def2
      .= (((f/.c))")" by A3,Def2
      .= (f|(dom (f^)))/.c by A1,A2,PARTFUN2:15;
  end;
  hence thesis by A1,PARTFUN2:1;
end;
