
theorem Th45:
  for f1,f2 be without+infty Function of [:NAT,NAT:],ExtREAL holds
    Partial_Sums_in_cod2(f1+f2)
      = Partial_Sums_in_cod2 f1 + Partial_Sums_in_cod2 f2 &
    Partial_Sums_in_cod1(f1+f2)
      = Partial_Sums_in_cod1 f1 + Partial_Sums_in_cod1 f2
proof
   let f1,f2 be without+infty Function of [:NAT,NAT:],ExtREAL;
   reconsider F1=-f1, F2=-f2 as without-infty Function of [:NAT,NAT:],ExtREAL;
   F1+F2 = -f1 -f2 by Th10 .= -(f1+f2) by Th9; then
A1:-(F1+F2) = f1+f2 by Th2; then
   Partial_Sums_in_cod2(f1+f2) = -(Partial_Sums_in_cod2(F1+F2)) by Th42
    .= -(Partial_Sums_in_cod2 F1 + Partial_Sums_in_cod2 F2) by Th44
    .= -( -(Partial_Sums_in_cod2 f1) + Partial_Sums_in_cod2 F2 ) by Th42
    .= -( -(Partial_Sums_in_cod2 f1) + -(Partial_Sums_in_cod2 f2) ) by Th42
    .= -(-(Partial_Sums_in_cod2 f1)) -(-(Partial_Sums_in_cod2 f2))
          by Th8
    .= Partial_Sums_in_cod2 f1 -(-(Partial_Sums_in_cod2 f2)) by Th2
    .= Partial_Sums_in_cod2 f1 + (-(-(Partial_Sums_in_cod2 f2))) by Th10;
   hence Partial_Sums_in_cod2(f1+f2)
      = Partial_Sums_in_cod2 f1 + Partial_Sums_in_cod2 f2 by Th2;
   Partial_Sums_in_cod1(f1+f2) = -(Partial_Sums_in_cod1(F1+F2)) by A1,Th42
    .= -(Partial_Sums_in_cod1 F1 + Partial_Sums_in_cod1 F2) by Th44
    .= -( -(Partial_Sums_in_cod1 f1) + Partial_Sums_in_cod1 F2 ) by Th42
    .= -( -(Partial_Sums_in_cod1 f1) + -(Partial_Sums_in_cod1 f2) ) by Th42
    .= -(-(Partial_Sums_in_cod1 f1)) -(-(Partial_Sums_in_cod1 f2))
          by Th8
    .= Partial_Sums_in_cod1 f1 -(-(Partial_Sums_in_cod1 f2)) by Th2
    .= Partial_Sums_in_cod1 f1 + (-(-(Partial_Sums_in_cod1 f2))) by Th10;
   hence thesis by Th2;
end;
