let X be non empty set ; :: thesis: for f being Functional_Sequence of X,REAL
for x being Element of X st x in dom (f . 0) holds
(superior_realsequence f) # x = superior_realsequence (R_EAL (f # x))
let f be Functional_Sequence of X,REAL; :: thesis: for x being Element of X st x in dom (f . 0) holds
(superior_realsequence f) # x = superior_realsequence (R_EAL (f # x))
let x be Element of X; :: thesis: ( x in dom (f . 0) implies (superior_realsequence f) # x = superior_realsequence (R_EAL (f # x)) )
set F = superior_realsequence f;
assume A1: x in dom (f . 0) ; :: thesis: (superior_realsequence f) # x = superior_realsequence (R_EAL (f # x))

now :: thesis: for n being Element of NAT holds ((superior_realsequence f) # x) . n = (superior_realsequence (R_EAL (f # x))) . n

let n be Element of NAT ; :: thesis: ((superior_realsequence f) # x) . n = (superior_realsequence (R_EAL (f # x))) . n
( dom ((superior_realsequence f) . n) = dom (f . 0) & ((superior_realsequence f) # x) . n = ((superior_realsequence f) . n) . x ) by Th5, MESFUNC5:def 13;
hence ((superior_realsequence f) # x) . n = (superior_realsequence (R_EAL (f # x))) . n by A1, Th5; :: thesis: verum

end;

hence (superior_realsequence f) # x = superior_realsequence (R_EAL (f # x)) by FUNCT_2:63; :: thesis: verum