let D be non empty set ; :: thesis: for r being Real
for H being Functional_Sequence of D,REAL
for X being set st X common_on_dom H holds
X common_on_dom r (#) H
let r be Real; :: thesis: for H being Functional_Sequence of D,REAL
for X being set st X common_on_dom H holds
X common_on_dom r (#) H
let H be Functional_Sequence of D,REAL ; :: thesis: for X being set st X common_on_dom H holds
X common_on_dom r (#) H
let X be set ; :: thesis: ( X common_on_dom H implies X common_on_dom r (#) H )
assume A1: X common_on_dom H ; :: thesis: X common_on_dom r (#) H

A2: now

let n be Element of NAT ; :: thesis: X c= dom ((r (#) H) . n)
dom (H . n) = dom (r (#) (H . n)) by VALUED_1:def 5
.= dom ((r (#) H) . n) by Def2 ;
hence X c= dom ((r (#) H) . n) by A1, Def10; :: thesis: verum

end;

X <> {} by A1, Def10;
hence X common_on_dom r (#) H by A2, Def10; :: thesis: verum