
theorem Th46:
  for f1 be without-infty Function of [:NAT,NAT:],ExtREAL,
      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 &
   Partial_Sums_in_cod2(f2-f1)
    = Partial_Sums_in_cod2 f2 - Partial_Sums_in_cod2 f1 &
   Partial_Sums_in_cod1(f2-f1)
    = Partial_Sums_in_cod1 f2 - Partial_Sums_in_cod1 f1
proof
   let f1 be without-infty Function of [:NAT,NAT:],ExtREAL,
       f2 be without+infty Function of [:NAT,NAT:],ExtREAL;
   Partial_Sums_in_cod2(f1-f2) = Partial_Sums_in_cod2(f1+(-f2)) by Th10
    .= Partial_Sums_in_cod2 f1 + Partial_Sums_in_cod2 (-f2)
       by Th44
    .= Partial_Sums_in_cod2 f1 + -(Partial_Sums_in_cod2 f2) by Th42;
   hence Partial_Sums_in_cod2(f1-f2)
     = Partial_Sums_in_cod2 f1 - Partial_Sums_in_cod2 f2 by Th10;
   Partial_Sums_in_cod1(f1-f2) = Partial_Sums_in_cod1(f1+(-f2)) by Th10
    .= Partial_Sums_in_cod1 f1 + Partial_Sums_in_cod1 (-f2)
       by Th44
    .= Partial_Sums_in_cod1 f1 + -(Partial_Sums_in_cod1 f2) by Th42;
   hence Partial_Sums_in_cod1(f1-f2)
     = Partial_Sums_in_cod1 f1 - Partial_Sums_in_cod1 f2 by Th10;
   Partial_Sums_in_cod2(f2-f1) = Partial_Sums_in_cod2(f2+(-f1)) by Th10
    .= Partial_Sums_in_cod2 f2 + Partial_Sums_in_cod2 (-f1)
       by Th45
    .= Partial_Sums_in_cod2 f2 + -(Partial_Sums_in_cod2 f1) by Th42;
   hence Partial_Sums_in_cod2(f2-f1)
     = Partial_Sums_in_cod2 f2 - Partial_Sums_in_cod2 f1 by Th10;
   Partial_Sums_in_cod1(f2-f1) = Partial_Sums_in_cod1(f2+(-f1)) by Th10
    .= Partial_Sums_in_cod1 f2 + Partial_Sums_in_cod1 (-f1)
       by Th45
    .= Partial_Sums_in_cod1 f2 + -(Partial_Sums_in_cod1 f1) by Th42;
   hence Partial_Sums_in_cod1(f2-f1)
     = Partial_Sums_in_cod1 f2 - Partial_Sums_in_cod1 f1 by Th10;
end;
