
theorem Th48:
for X be non empty set, F be Functional_Sequence of X,ExtREAL,
  x be Element of X st F#x is summable holds
    (-F)#x is summable & Sum((-F)#x) = -Sum(F#x)
proof
    let X be non empty set, F be Functional_Sequence of X,ExtREAL,
    x be Element of X;
    assume A1: F#x is summable; then
    -(F#x) is summable by Th45;
    hence (-F)#x is summable by Th38;
A2: -(F#x) = (-F)#x by Th38;
    Partial_Sums(F#x) is convergent by A1,MESFUNC9:def 2; then
    lim -(Partial_Sums(F#x)) = - lim Partial_Sums(F#x) by DBLSEQ_3:17; then
    lim Partial_Sums(-(F#x)) = - lim Partial_Sums(F#x) by Th44; then
    lim Partial_Sums((-F)#x) = - Sum (F#x) by A2,MESFUNC9:def 3;
    hence Sum((-F)#x) = -Sum(F#x) by MESFUNC9:def 3;
end;
