let X be non empty set ; :: thesis:
let S be SigmaField of X; :: thesis:
let f be with_the_same_dom Functional_Sequence of X,COMPLEX; :: thesis:
let g be PartFunc of X,COMPLEX; :: thesis:
let E be Element of S; :: thesis:
assume that
A1: dom (f . 0) = E and
A2: for n being Nat holds f . n is E -measurable and
A3: dom g = E and
A4: for x being Element of X st x in E holds
( f # x is convergent & g . x = lim (f # x) ) ; :: thesis:

A5: now :: thesis:

let n be Nat; :: thesis:
f . n is E -measurable by A2;
then Im (f . n) is E -measurable by MESFUN6C:def 1;
hence (Im f) . n is E -measurable by Th24; :: thesis:

end;

A6: dom (Im g) = E by A3, COMSEQ_3:def 4;

A7: now :: thesis:

let x be Element of X; :: thesis:
assume A8: x in E ; :: thesis:
then A9: f # x is convergent by A4;
then Im (f # x) is convergent ;
hence (Im f) # x is convergent by A1, A8, Th23; :: thesis:
g . x = lim (f # x) by A4, A8;
then Im (g . x) = lim (Im (f # x)) by A9, COMSEQ_3:41;
then Im (g . x) = lim ((Im f) # x) by A1, A8, Th23;
hence (Im g) . x = lim ((Im f) # x) by A6, A8, COMSEQ_3:def 4; :: thesis:

end;

dom ((Im f) . 0) = E by A1, Def12;
then R_EAL (Im g) is E -measurable by A5, A6, A7, Th22;
then A10: Im g is E -measurable by MESFUNC6:def 1;

A11: now :: thesis:

let n be Nat; :: thesis:
f . n is E -measurable by A2;
then Re (f . n) is E -measurable by MESFUN6C:def 1;
hence (Re f) . n is E -measurable by Th24; :: thesis:

end;

A12: dom (Re g) = E by A3, COMSEQ_3:def 3;

A13: now :: thesis:

let x be Element of X; :: thesis:
assume A14: x in E ; :: thesis:
then A15: f # x is convergent by A4;
then Re (f # x) is convergent ;
hence (Re f) # x is convergent by A1, A14, Th23; :: thesis:
g . x = lim (f # x) by A4, A14;
then Re (g . x) = lim (Re (f # x)) by A15, COMSEQ_3:41;
then Re (g . x) = lim ((Re f) # x) by A1, A14, Th23;
hence (Re g) . x = lim ((Re f) # x) by A12, A14, COMSEQ_3:def 3; :: thesis:

end;

dom ((Re f) . 0) = E by A1, Def11;
then R_EAL (Re g) is E -measurable by A11, A12, A13, Th22;
then Re g is E -measurable by MESFUNC6:def 1;
hence g is E -measurable by A10, MESFUN6C:def 1; :: thesis: