let f be complex-valued Function; :: thesis: (f ^) ^ = f | (dom (f ^))
A1: dom ((f ^) ^) = dom (f | (dom (f ^))) by Th6;

now :: thesis: for c being object st c in dom ((f ^) ^) holds
((f ^) ^) . c = (f | (dom (f ^))) . c

let c be object ; :: thesis: ( c in dom ((f ^) ^) implies ((f ^) ^) . c = (f | (dom (f ^))) . c )
assume A2: c in dom ((f ^) ^) ; :: thesis: ((f ^) ^) . c = (f | (dom (f ^))) . c
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, FUNCT_1:47 ; :: thesis: verum

end;

hence (f ^) ^ = f | (dom (f ^)) by A1, FUNCT_1:2; :: thesis: verum