let f, g be Function; :: thesis: for S being non void Signature
for E being Extension of S st f,g form_a_replacement_in E holds
E with-replacement f,g is Extension of S with-replacement f,g
let S be non void Signature; :: thesis: for E being Extension of S st f,g form_a_replacement_in E holds
E with-replacement f,g is Extension of S with-replacement f,g
let E be Extension of S; :: thesis: ( f,g form_a_replacement_in E implies E with-replacement f,g is Extension of S with-replacement f,g )
set f9 = the carrier of E -indexing f;
set g9 = the carrier' of E -indexing g;
set gg = the carrier' of S -indexing g;
set T = E with-replacement f,g;
A1: the carrier' of S -indexing (the carrier' of S -indexing g) = the carrier' of S -indexing g by Th11;
assume A2: f,g form_a_replacement_in E ; :: thesis: E with-replacement f,g is Extension of S with-replacement f,g
then f,g form_a_replacement_in S by Th54;
then the carrier of S -indexing f,the carrier' of S -indexing g form_a_replacement_in S by Th31;
then A3: f,the carrier' of S -indexing g form_a_replacement_in S by A1, Th31;
A4: S is Subsignature of E by Def5;
then A5: (the carrier' of E -indexing g) | the carrier' of S = the carrier' of S -indexing g by Th18, INSTALG1:11;
the carrier of E -indexing f,the carrier' of E -indexing g form_morphism_between E,E with-replacement f,g by A2, Th41;
then A6: S with-replacement ((the carrier of E -indexing f) | the carrier of S),((the carrier' of E -indexing g) | the carrier' of S) is Subsignature of E with-replacement f,g by A4, Th40, INSTALG1:19;
(the carrier of E -indexing f) | the carrier of S = the carrier of S -indexing f by A4, Th18, INSTALG1:11;
then S with-replacement ((the carrier of E -indexing f) | the carrier of S),((the carrier' of E -indexing g) | the carrier' of S) = S with-replacement f,(the carrier' of S -indexing g) by A3, A5, Th44;
hence S with-replacement f,g is Subsignature of E with-replacement f,g by A2, A6, Th45, Th54; :: according to ALGSPEC1:def 5 :: thesis: verum