let f, g, h be Function; :: thesis: for A being set

for F being Function st f | A = g | A holds

(F .: (h,f)) | A = (F .: (h,g)) | A

let A be set ; :: thesis: for F being Function st f | A = g | A holds

(F .: (h,f)) | A = (F .: (h,g)) | A

let F be Function; :: thesis: ( f | A = g | A implies (F .: (h,f)) | A = (F .: (h,g)) | A )

assume A1: f | A = g | A ; :: thesis: (F .: (h,f)) | A = (F .: (h,g)) | A

thus (F .: (h,f)) | A = F * (<:h,f:> | A) by RELAT_1:83

.= F * <:h,(f | A):> by Th6

.= F * (<:h,g:> | A) by A1, Th6

.= (F .: (h,g)) | A by RELAT_1:83 ; :: thesis: verum

for F being Function st f | A = g | A holds

(F .: (h,f)) | A = (F .: (h,g)) | A

let A be set ; :: thesis: for F being Function st f | A = g | A holds

(F .: (h,f)) | A = (F .: (h,g)) | A

let F be Function; :: thesis: ( f | A = g | A implies (F .: (h,f)) | A = (F .: (h,g)) | A )

assume A1: f | A = g | A ; :: thesis: (F .: (h,f)) | A = (F .: (h,g)) | A

thus (F .: (h,f)) | A = F * (<:h,f:> | A) by RELAT_1:83

.= F * <:h,(f | A):> by Th6

.= F * (<:h,g:> | A) by A1, Th6

.= (F .: (h,g)) | A by RELAT_1:83 ; :: thesis: verum