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