let UA be Universal_Algebra; :: thesis: for h being Function st dom h = UAEnd UA & ( for x being object st x in UAEnd UA holds
h . x = 0 .--> x ) holds
rng h = MSAEnd (MSAlg UA)
let h be Function; :: thesis: ( dom h = UAEnd UA & ( for x being object st x in UAEnd UA holds
h . x = 0 .--> x ) implies rng h = MSAEnd (MSAlg UA) )
assume that
A1: dom h = UAEnd UA and
A2: for x being object st x in UAEnd UA holds
h . x = 0 .--> x ; :: thesis: rng h = MSAEnd (MSAlg UA)
A3: MSAEnd (MSAlg UA) c= rng h

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in MSAEnd (MSAlg UA) or x in rng h )
assume A4: x in MSAEnd (MSAlg UA) ; :: thesis: x in rng h
then reconsider f = x as ManySortedFunction of (MSAlg UA),(MSAlg UA) by Def4;
the carrier of (MSSign UA) = {0} by MSUALG_1:def 8;
then A5: f = 0 .--> (f . 0) by AUTALG_1:11;
A6: f is_homomorphism MSAlg UA, MSAlg UA by A4, Def4;
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 AUTALG_1:31;
MSAlg q9 = f by A5, MSUHOM_1:def 3;
then MSAlg q9 is_homomorphism MSAlg UA,(MSAlg UA) Over (MSSign UA) by A6, MSUHOM_1:9;
then q9 is_homomorphism by MSUHOM_1:21;
hence q in dom h by A1, Def1; :: thesis: x = h . q
hence x = h . q by A1, A2, A5; :: thesis: verum

end;

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

end;

rng h c= MSAEnd (MSAlg UA)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng h or x in MSAEnd (MSAlg UA) )
assume x in rng h ; :: thesis: x in MSAEnd (MSAlg UA)
then consider q being object such that
A7: q in dom h and
A8: x = h . q by FUNCT_1:def 3;
consider q9 being Element of UAEnd UA such that
A9: q9 = q by A1, A7;
( x = 0 .--> q & 0 .--> q is ManySortedFunction of (MSAlg UA),(MSAlg UA) ) by A1, A2, A7, A8, Th12;
then consider d being ManySortedFunction of (MSAlg UA),(MSAlg UA) such that
A10: d = x ;
q9 is_homomorphism by Def1;
then A11: MSAlg q9 is_homomorphism MSAlg UA,(MSAlg UA) Over (MSSign UA) by MSUHOM_1:16;
MSAlg q9 = 0 .--> q by A9, MSUHOM_1:def 3
.= x by A1, A2, A7, A8 ;
then d is_homomorphism MSAlg UA, MSAlg UA by A10, A11, MSUHOM_1:9;
hence x in MSAEnd (MSAlg UA) by A10, Def4; :: thesis: verum

end;

hence rng h = MSAEnd (MSAlg UA) by A3, XBOOLE_0:def 10; :: thesis: verum