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

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 )

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 MSAAut (MSAlg UA) or x in rng h )
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;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

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

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