let S, S9 be non empty non void ManySortedSign ; :: thesis: for A being non-empty MSAlgebra of S
for f being Function of the carrier of S9,the carrier of S
for g being Function st f,g form_morphism_between S9,S holds
for B being non-empty MSAlgebra of S9 st B = A | S9,f,g holds
for s1, s2 being SortSymbol of S9
for t being Function st t is_e.translation_of B,s1,s2 holds
t is_e.translation_of A,f . s1,f . s2
let A be non-empty MSAlgebra of S; :: thesis: for f being Function of the carrier of S9,the carrier of S
for g being Function st f,g form_morphism_between S9,S holds
for B being non-empty MSAlgebra of S9 st B = A | S9,f,g holds
for s1, s2 being SortSymbol of S9
for t being Function st t is_e.translation_of B,s1,s2 holds
t is_e.translation_of A,f . s1,f . s2
let f be Function of the carrier of S9,the carrier of S; :: thesis: for g being Function st f,g form_morphism_between S9,S holds
for B being non-empty MSAlgebra of S9 st B = A | S9,f,g holds
for s1, s2 being SortSymbol of S9
for t being Function st t is_e.translation_of B,s1,s2 holds
t is_e.translation_of A,f . s1,f . s2
let g be Function; :: thesis: ( f,g form_morphism_between S9,S implies for B being non-empty MSAlgebra of S9 st B = A | S9,f,g holds
for s1, s2 being SortSymbol of S9
for t being Function st t is_e.translation_of B,s1,s2 holds
t is_e.translation_of A,f . s1,f . s2 )
assume A1: f,g form_morphism_between S9,S ; :: thesis: for B being non-empty MSAlgebra of S9 st B = A | S9,f,g holds
for s1, s2 being SortSymbol of S9
for t being Function st t is_e.translation_of B,s1,s2 holds
t is_e.translation_of A,f . s1,f . s2
A2: ( dom g = the carrier' of S9 & rng g c= the carrier' of S ) by A1, PUA2MSS1:def 13;
let B be non-empty MSAlgebra of S9; :: thesis: ( B = A | S9,f,g implies for s1, s2 being SortSymbol of S9
for t being Function st t is_e.translation_of B,s1,s2 holds
t is_e.translation_of A,f . s1,f . s2 )
assume A3: B = A | S9,f,g ; :: thesis: for s1, s2 being SortSymbol of S9
for t being Function st t is_e.translation_of B,s1,s2 holds
t is_e.translation_of A,f . s1,f . s2
reconsider g = g as Function of the carrier' of S9,the carrier' of S by A2, FUNCT_2:def 1, RELSET_1:11;
let s1, s2 be SortSymbol of S9; :: thesis: for t being Function st t is_e.translation_of B,s1,s2 holds
t is_e.translation_of A,f . s1,f . s2
let t be Function; :: thesis: ( t is_e.translation_of B,s1,s2 implies t is_e.translation_of A,f . s1,f . s2 )
given o being OperSymbol of S9 such that A4: the_result_sort_of o = s2 and
A5: ex i being Element of NAT st
( i in dom (the_arity_of o) & (the_arity_of o) /. i = s1 & ex a being Function st
( a in Args o,B & t = transl o,i,a,B ) ) ; :: according to MSUALG_6:def 5 :: thesis: t is_e.translation_of A,f . s1,f . s2
A6: f * (the_arity_of o) = the_arity_of (g . o) by A1, PUA2MSS1:def 13;
take g . o ; :: according to MSUALG_6:def 5 :: thesis: ( the_result_sort_of (g . o) = f . s2 & ex b₁ being Element of NAT st
( b₁ in dom (the_arity_of (g . o)) & (the_arity_of (g . o)) /. b₁ = f . s1 & ex b₂ being set st
( b₂ in Args (g . o),A & t = transl (g . o),b₁,b₂,A ) ) )
f * the ResultSort of S9 = the ResultSort of S * g by A1, PUA2MSS1:def 13;
hence the_result_sort_of (g . o) = (f * the ResultSort of S9) . o by FUNCT_2:21
.= f . s2 by A4, FUNCT_2:21 ;
:: thesis: ex b₁ being Element of NAT st
( b₁ in dom (the_arity_of (g . o)) & (the_arity_of (g . o)) /. b₁ = f . s1 & ex b₂ being set st
( b₂ in Args (g . o),A & t = transl (g . o),b₁,b₂,A ) )
consider i being Element of NAT , a being Function such that
A7: i in dom (the_arity_of o) and
A8: (the_arity_of o) /. i = s1 and
A9: a in Args o,B and
A10: t = transl o,i,a,B by A5;
take i ; :: thesis: ( i in dom (the_arity_of (g . o)) & (the_arity_of (g . o)) /. i = f . s1 & ex b₁ being set st
( b₁ in Args (g . o),A & t = transl (g . o),i,b₁,A ) )
( rng (the_arity_of o) c= the carrier of S9 & dom f = the carrier of S9 ) by FINSEQ_1:def 4, FUNCT_2:def 1;
hence i in dom (the_arity_of (g . o)) by A7, A6, RELAT_1:46; :: thesis: ( (the_arity_of (g . o)) /. i = f . s1 & ex b₁ being set st
( b₁ in Args (g . o),A & t = transl (g . o),i,b₁,A ) )
hence A11: (the_arity_of (g . o)) /. i = (the_arity_of (g . o)) . i by PARTFUN1:def 8
.= f . ((the_arity_of o) . i) by A7, A6, FUNCT_1:23
.= f . s1 by A7, A8, PARTFUN1:def 8 ;
:: thesis: ex b₁ being set st
( b₁ in Args (g . o),A & t = transl (g . o),i,b₁,A )
then A12: dom (transl (g . o),i,a,A) = the Sorts of A . (f . s1) by MSUALG_6:def 4;
A13: the Sorts of B = the Sorts of A * f by A1, A3, Def3;
then A14: the Sorts of B . s1 = the Sorts of A . (f . s1) by FUNCT_2:21;

A15: now

let x be set ; :: thesis: ( x in the Sorts of B . s1 implies t . x = (transl (g . o),i,a,A) . x )
assume x in the Sorts of B . s1 ; :: thesis: t . x = (transl (g . o),i,a,A) . x
then ( t . x = (Den o,B) . (a +* i,x) & (transl (g . o),i,a,A) . x = (Den (g . o),A) . (a +* i,x) ) by A8, A10, A11, A14, MSUALG_6:def 4;
hence t . x = (transl (g . o),i,a,A) . x by A1, A3, Th24; :: thesis: verum

end;

take a ; :: thesis: ( a in Args (g . o),A & t = transl (g . o),i,a,A )
thus a in Args (g . o),A by A1, A3, A9, Th25; :: thesis: t = transl (g . o),i,a,A
dom t = the Sorts of B . s1 by A8, A10, MSUALG_6:def 4;
hence t = transl (g . o),i,a,A by A12, A13, A15, FUNCT_1:9, FUNCT_2:21; :: thesis: verum