let f, h, F be Function; :: thesis: for x being set holds (F [:] (f,x)) * h = F [:] ((f * h),x)
let x be set ; :: thesis: (F [:] (f,x)) * h = F [:] ((f * h),x)
A1: dom ((dom f) --> x) = dom f by Th19;
then A2: dom (((dom f) --> x) * h) = dom (f * h) by RELAT_1:198;

A3: now

let z be set ; :: thesis: ( z in dom (((dom f) --> x) * h) implies (((dom f) --> x) * h) . z = x )
assume A4: z in dom (((dom f) --> x) * h) ; :: thesis: (((dom f) --> x) * h) . z = x
then A5: h . z in dom ((dom f) --> x) by FUNCT_1:21;
thus (((dom f) --> x) * h) . z = ((dom f) --> x) . (h . z) by A4, FUNCT_1:22
.= x by A1, A5, Th13 ; :: thesis: verum

end;

thus (F [:] (f,x)) * h = (F .: (f,((dom f) --> x))) * h
.= F .: ((f * h),(((dom f) --> x) * h)) by Th31
.= F [:] ((f * h),x) by A2, A3, Th17 ; :: thesis: verum