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
(inferior_realsequence f) # x = inferior_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
(inferior_realsequence f) # x = inferior_realsequence (R_EAL (f # x))
let x be Element of X; :: thesis: ( x in dom (f . 0) implies (inferior_realsequence f) # x = inferior_realsequence (R_EAL (f # x)) )
set F = inferior_realsequence f;
assume A1: x in dom (f . 0) ; :: thesis: (inferior_realsequence f) # x = inferior_realsequence (R_EAL (f # x))

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

let n be Element of NAT ; :: thesis: ((inferior_realsequence f) # x) . n = (inferior_realsequence (R_EAL (f # x))) . n
( dom ((inferior_realsequence f) . n) = dom (f . 0) & ((inferior_realsequence f) # x) . n = ((inferior_realsequence f) . n) . x ) by Th4, MESFUNC5:def 13;
hence ((inferior_realsequence f) # x) . n = (inferior_realsequence (R_EAL (f # x))) . n by A1, Th4; :: thesis: verum

end;

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