
theorem Th49:
for X be non empty set, S be SigmaField of X,
 F be Functional_Sequence of X,ExtREAL st
  F is additive & F is with_the_same_dom &
  (for x be Element of X st x in dom(F.0) holds F#x is summable)
  holds lim Partial_Sums (-F) = -(lim Partial_Sums F)
proof
    let X be non empty set, S be SigmaField of X,
    F be Functional_Sequence of X,ExtREAL;
    assume that
A1:  F is additive and
A2:  F is with_the_same_dom and
A3:  for x be Element of X st x in dom(F.0) holds F#x is summable;
    set G = -F;
    for n be Element of NAT holds (Partial_Sums G).n = (-(Partial_Sums F)).n
      by Th42; then
A4: Partial_Sums G = -(Partial_Sums F) by FUNCT_2:def 7;
A5: dom(lim Partial_Sums G) = dom((Partial_Sums G).0) by MESFUNC8:def 9
     .= dom(G.0) by MESFUNC9:def 4 .= dom(-(F.0)) by Th37
     .= dom(F.0) by MESFUNC1:def 7;
A6: dom(-(lim Partial_Sums F)) = dom(lim Partial_Sums F) by MESFUNC1:def 7;
    then
A7: dom(-(lim Partial_Sums F)) = dom((Partial_Sums F).0) by MESFUNC8:def 9
     .= dom(F.0) by MESFUNC9:def 4;
    for x be Element of X st x in dom(lim Partial_Sums G) holds
    (lim Partial_Sums G).x = (-(lim Partial_Sums F)).x
    proof
     let x be Element of X;
     assume A8: x in dom(lim Partial_Sums G); then
     F#x is summable by A3,A5; then
     Partial_Sums(F#x) is convergent by MESFUNC9:def 2; then
A9:  (Partial_Sums F)#x is convergent by A1,A2,A8,A5,MESFUNC9:33;
     (Partial_Sums G)#x = -((Partial_Sums F)#x) by A4,Th38; then
A10: lim((Partial_Sums G)#x) = - lim((Partial_Sums F)#x)
       by A9,DBLSEQ_3:17;
     (-(lim Partial_Sums F)).x = -( (lim Partial_Sums F).x)
       by A7,A5,A8,MESFUNC1:def 7; then
     (-(lim Partial_Sums F)).x = -( lim((Partial_Sums F)#x) )
       by A7,A5,A8,A6,MESFUNC8:def 9;
     hence (lim Partial_Sums G).x = (-(lim Partial_Sums F)).x
       by A8,A10,MESFUNC8:def 9;
    end;
    hence lim Partial_Sums G = -(lim Partial_Sums F) by A7,A5,PARTFUN1:5;
end;
