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
f,g form_a_replacement_in S
let S be non void Signature; :: thesis: for E being Extension of S st f,g form_a_replacement_in E holds
f,g form_a_replacement_in S
let E be Extension of S; :: thesis: ( f,g form_a_replacement_in E implies f,g form_a_replacement_in S )
set f' = the carrier of E -indexing f;
set g' = the carrier' of E -indexing g;
set T = E with-replacement f,g;
assume f,g form_a_replacement_in E ; :: thesis: f,g form_a_replacement_in S
then A1: ( the carrier of E -indexing f,the carrier' of E -indexing g form_morphism_between E,E with-replacement f,g & S is Subsignature of E ) by Def5, Th41;
then A2: (the carrier of E -indexing f) | the carrier of S,(the carrier' of E -indexing g) | the carrier' of S form_a_replacement_in S by Th32, INSTALG1:19;
A3: (the carrier of E -indexing f) | the carrier of S = the carrier of S -indexing f by Th18, A1, INSTALG1:11;
(the carrier' of E -indexing g) | the carrier' of S = the carrier' of S -indexing g by Th18, A1, INSTALG1:11;
hence f,g form_a_replacement_in S by A2, A3, Th31; :: thesis: verum