let I be set ; :: thesis: for A being ManySortedSet of I
for B being non-empty ManySortedSet of I
for F being ManySortedFunction of A,B
for X being ManySortedSubset of A holds doms (F || X) c= doms F
let A be ManySortedSet of I; :: thesis: for B being non-empty ManySortedSet of I
for F being ManySortedFunction of A,B
for X being ManySortedSubset of A holds doms (F || X) c= doms F
let B be non-empty ManySortedSet of I; :: thesis: for F being ManySortedFunction of A,B
for X being ManySortedSubset of A holds doms (F || X) c= doms F
let F be ManySortedFunction of A,B; :: thesis: for X being ManySortedSubset of A holds doms (F || X) c= doms F
let X be ManySortedSubset of A; :: thesis: doms (F || X) c= doms F
let i be object ; :: according to PBOOLE:def 2 :: thesis: ( not i in I or (doms (F || X)) . i c= (doms F) . i )
A1: dom (F || X) = I by PARTFUN1:def 2;
assume A2: i in I ; :: thesis: (doms (F || X)) . i c= (doms F) . i
then reconsider f = F . i as Function of (A . i),(B . i) by PBOOLE:def 15;
dom F = I by PARTFUN1:def 2;
then A3: (doms F) . i = dom f by A2, FUNCT_6:22;
(F || X) . i = f | (X . i) by A2, MSAFREE:def 1;
then (doms (F || X)) . i = dom (f | (X . i)) by A2, A1, FUNCT_6:22;
hence (doms (F || X)) . i c= (doms F) . i by A3, RELAT_1:60; :: thesis: verum