let UA be Universal_Algebra; :: thesis: for h being Function st dom h = UAAut UA & ( for x being object st x in UAAut UA holds
h . x = 0 .--> x ) holds
rng h = MSAAut (MSAlg UA)
let h be Function; :: thesis: ( dom h = UAAut UA & ( for x being object st x in UAAut UA holds
h . x = 0 .--> x ) implies rng h = MSAAut (MSAlg UA) )
assume that
A1: dom h = UAAut UA and
A2: for x being object st x in UAAut UA holds
h . x = 0 .--> x ; :: thesis: rng h = MSAAut (MSAlg UA)
thus rng h c= MSAAut (MSAlg UA) :: according to XBOOLE_0:def 10 :: thesis: MSAAut (MSAlg UA) c= rng h

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng h or x in MSAAut (MSAlg UA) )
assume x in rng h ; :: thesis: x in MSAAut (MSAlg UA)
then consider q being object such that
A3: q in dom h and
A4: x = h . q by FUNCT_1:def 3;
consider q9 being Element of UAAut UA such that
A5: q9 = q by A1, A3;
( x = 0 .--> q & 0 .--> q is ManySortedFunction of (MSAlg UA),(MSAlg UA) ) by A1, A2, A3, A4, Th32;
then consider d being ManySortedFunction of (MSAlg UA),(MSAlg UA) such that
A6: d = x ;
q9 is_isomorphism by Def1;
then A7: MSAlg q9 is_isomorphism MSAlg UA,(MSAlg UA) Over (MSSign UA) by MSUHOM_1:20;
MSAlg q9 = 0 .--> q by A5, MSUHOM_1:def 3
.= x by A1, A2, A3, A4 ;
then d is_isomorphism MSAlg UA, MSAlg UA by A6, A7, MSUHOM_1:9;
hence x in MSAAut (MSAlg UA) by A6, Def5; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in MSAAut (MSAlg UA) or x in rng h )
assume A8: x in MSAAut (MSAlg UA) ; :: thesis: x in rng h
then reconsider f = x as ManySortedFunction of (MSAlg UA),(MSAlg UA) by Th19;
the carrier of (MSSign UA) = {0} by MSUALG_1:def 8;
then A9: f = 0 .--> (f . 0) by Th11;
A10: f is_isomorphism MSAlg UA, MSAlg UA by A8, Def5;
ex q being set st
( q in dom h & x = h . q )

proof

take q = f . 0; :: thesis: ( q in dom h & x = h . q )
f is ManySortedFunction of (MSAlg UA),((MSAlg UA) Over (MSSign UA)) by MSUHOM_1:9;
then reconsider q9 = q as Function of UA,UA by Th31;
MSAlg q9 = f by A9, MSUHOM_1:def 3;
then MSAlg q9 is_isomorphism MSAlg UA,(MSAlg UA) Over (MSSign UA) by A10, MSUHOM_1:9;
then q9 is_isomorphism by MSUHOM_1:24;
hence q in dom h by A1, Def1; :: thesis: x = h . q
hence x = h . q by A1, A2, A9; :: thesis: verum

end;

hence x in rng h by FUNCT_1:def 3; :: thesis: verum