let f, g be Function; :: thesis: for A being set
for F being Function
for x being object st f | A = g | A holds
(F [:] (f,x)) | A = (F [:] (g,x)) | A
let A be set ; :: thesis: for F being Function
for x being object st f | A = g | A holds
(F [:] (f,x)) | A = (F [:] (g,x)) | A
let F be Function; :: thesis: for x being object st f | A = g | A holds
(F [:] (f,x)) | A = (F [:] (g,x)) | A
let x be object ; :: thesis: ( f | A = g | A implies (F [:] (f,x)) | A = (F [:] (g,x)) | A )
assume A1: f | A = g | A ; :: thesis: (F [:] (f,x)) | A = (F [:] (g,x)) | A
(dom f) /\ A = dom (f | A) by RELAT_1:61
.= (dom g) /\ A by A1, RELAT_1:61 ;
then A2: ((dom f) --> x) | A = ((dom g) /\ A) --> x by Th12
.= ((dom g) --> x) | A by Th12 ;
thus (F [:] (f,x)) | A = (F .: (f,((dom f) --> x))) | A
.= (F .: (g,((dom f) --> x))) | A by A1, Th23
.= (F .: (g,((dom g) --> x))) | A by A2, Th24
.= (F [:] (g,x)) | A ; :: thesis: verum