
theorem
for f1 be without-infty Function of [:NAT,NAT:],ExtREAL,
    f2 be without+infty Function of [:NAT,NAT:],ExtREAL holds
  Partial_Sums(f1 - f2) = Partial_Sums f1 - Partial_Sums f2
& Partial_Sums(f2 - f1) = Partial_Sums f2 - Partial_Sums f1
proof
   let f1 be without-infty Function of [:NAT,NAT:],ExtREAL,
       f2 be without+infty Function of [:NAT,NAT:],ExtREAL;
   Partial_Sums(f1-f2)
    = Partial_Sums_in_cod2(Partial_Sums_in_cod1 f1 - Partial_Sums_in_cod1 f2)
      by Th46;
   hence Partial_Sums(f1-f2) = Partial_Sums f1 - Partial_Sums f2 by Th46;
   Partial_Sums(f2-f1)
    = Partial_Sums_in_cod2(Partial_Sums_in_cod1 f2 - Partial_Sums_in_cod1 f1)
      by Th46;
   hence Partial_Sums(f2-f1) = Partial_Sums f2 - Partial_Sums f1 by Th46;
end;
