
theorem Th50:
for X be non empty set, S be SigmaField of X,
 F,G be Functional_Sequence of X,ExtREAL, E be Element of S
 st E c= dom(F.0) & F is additive & F is with_the_same_dom
  & (for n be Nat holds G.n = (F.n)|E)
 holds lim(Partial_Sums G) = (lim(Partial_Sums F))|E
proof
    let X be non empty set, S be SigmaField of X,
    F,G be Functional_Sequence of X,ExtREAL, E be Element of S;
    assume that
A1:  E c= dom(F.0) and
A2:  F is additive and
A3:  F is with_the_same_dom and
A5:  for n be Nat holds G.n = (F.n)|E;
A6: G is additive with_the_same_dom Functional_Sequence of X,ExtREAL
      by A2,A3,A5,MESFUNC9:18,31;
  dom((F.0)|E) = E by A1,RELAT_1:62; then
A8: E = dom(G.0) by A5;
A9: for x be Element of X st x in E holds F#x = G#x
  proof
    let x be Element of X;
    assume
A10: x in E;
    for n be Element of NAT holds (F#x).n = (G#x).n
    proof
      let n be Element of NAT;
      dom(G.n) = E by A8,MESFUNC8:def 2,A3,A5,MESFUNC9:18;
      then x in dom((F.n)|E) by A5,A10;
      then ((F.n)|E).x = (F.n).x by FUNCT_1:47;
      then
A11:  (G.n).x = (F.n).x by A5;
      (F#x).n = (F.n).x by MESFUNC5:def 13;
      hence thesis by A11,MESFUNC5:def 13;
    end;
    hence thesis by FUNCT_2:def 7;
  end;
    set E1 = dom(F.0);
    set PF = Partial_Sums F;
    set PG = Partial_Sums G;
A13: dom(lim PG) = dom(PG.0) by MESFUNC8:def 9;
    dom(PF.0) = E1 by A2,A3,MESFUNC9:29;
    then
A14: E c= dom(lim(PF)) by A1,MESFUNC8:def 9;
A15: for x being Element of X st x in dom(lim PG) holds (lim PG).x = (lim PF).x
    proof
      let x be Element of X;
      set PFx = Partial_Sums(F#x);
      set PGx = Partial_Sums(G#x);
      assume
A16:  x in dom(lim PG); then
      x in dom(G.0) by A6,A13,MESFUNC9:29; then
      x in dom((F.0)|E) by A5;
      then
A17:  x in E by A1,RELAT_1:62;
      for n be Element of NAT holds (PG#x).n = (PF#x).n
      proof
        let n be Element of NAT;
A18:    PGx.n = (PG#x).n by A6,A8,A17,MESFUNC9:32;
        PFx.n = (PF#x).n by A1,A2,A3,A17,MESFUNC9:32;
        hence thesis by A9,A17,A18;
      end;
      then
A19:  lim(PG#x) = lim(PF#x) by FUNCT_2:63;
      (lim PG).x = lim(PG#x) by A16,MESFUNC8:def 9;
      hence thesis by A14,A17,A19,MESFUNC8:def 9;
    end;
A20: dom(PG.0) = dom(G.0) by A6,MESFUNC9:29;
A22: dom((lim PF)|E) = E by A14,RELAT_1:62;
    for x be Element of X st x in dom((lim PG)|E) holds ((lim PG)|E).x =
    ((lim PF)|E).x
    proof
      let x be Element of X;
      assume
A24:  x in dom((lim PG)|E);
      then ((lim PF)|E).x = (lim PF).x by A8,A13,A20,A22,FUNCT_1:47;
      hence thesis by A8,A13,A20,A15,A24;
    end;
    hence thesis by A8,A20,A13,A22,PARTFUN1:5;
end;
