
theorem
  for f be without+infty Function of [:NAT,NAT:],ExtREAL,
      n,m be Nat holds
   (Partial_Sums f).(n+1,m)
     = (Partial_Sums_in_cod2 f).(n+1,m) + (Partial_Sums f).(n,m) &
   (Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)).(n,m+1)
     = (Partial_Sums_in_cod1 f).(n,m+1)
       + (Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)).(n,m)
proof
   let f be without+infty Function of [:NAT,NAT:],ExtREAL;
   let n,m be Nat;
   reconsider g = -f as without-infty Function of [:NAT,NAT:],ExtREAL;
A2:dom(-(Partial_Sums_in_cod2(Partial_Sums_in_cod1 g))) = [:NAT,NAT:]
 & dom(-(Partial_Sums_in_cod1(Partial_Sums_in_cod2 g))) = [:NAT,NAT:]
      by FUNCT_2:def 1;
A4:dom(-(Partial_Sums_in_cod2 g)) = [:NAT,NAT:]
 & dom(-(Partial_Sums_in_cod1 g)) = [:NAT,NAT:] by FUNCT_2:def 1;
A5:dom(-(Partial_Sums g)) = [:NAT,NAT:] by FUNCT_2:def 1;
   n in NAT & m in NAT by ORDINAL1:def 12; then
A3:[n+1,m] in [:NAT,NAT:] & [n,m] in [:NAT,NAT:]
 & [n,m+1] in [:NAT,NAT:] by ZFMISC_1:87;
A1:Partial_Sums f = Partial_Sums_in_cod2(Partial_Sums_in_cod1 (-(g))) by Th2
    .= Partial_Sums_in_cod2(-(Partial_Sums_in_cod1 g)) by Th42
    .= -(Partial_Sums g) by Th42;
   thus (Partial_Sums f).(n+1,m)
    = -( (Partial_Sums_in_cod2(Partial_Sums_in_cod1 g)).(n+1,m) )
          by A1,A2,A3,MESFUNC1:def 7
   .= -( (Partial_Sums_in_cod2 g).(n+1,m) + (Partial_Sums g).(n,m) ) by Th47
   .= - (Partial_Sums_in_cod2 g).(n+1,m) - (Partial_Sums g).(n,m)
          by XXREAL_3:25
   .= (-(Partial_Sums_in_cod2 g)).(n+1,m) + -(Partial_Sums g).(n,m)
          by A3,A4,MESFUNC1:def 7
   .= (Partial_Sums_in_cod2(-g)).(n+1,m) + -(Partial_Sums g).(n,m) by Th42
   .= (Partial_Sums_in_cod2 f).(n+1,m) + -(Partial_Sums g).(n,m) by Th2
   .= (Partial_Sums_in_cod2 f).(n+1,m) + (Partial_Sums f).(n,m)
          by A1,A3,A5,MESFUNC1:def 7;
   thus (Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)).(n,m+1)
    = (Partial_Sums_in_cod1( (Partial_Sums_in_cod2(-g)) )).(n,m+1) by Th2
   .= (Partial_Sums_in_cod1( -(Partial_Sums_in_cod2 g) )).(n,m+1) by Th42
   .= ( -(Partial_Sums_in_cod1(Partial_Sums_in_cod2 g)) ).(n,m+1) by Th42
   .= -( (Partial_Sums_in_cod1(Partial_Sums_in_cod2 g)).(n,m+1) )
         by A3,A2,MESFUNC1:def 7
   .= -( (Partial_Sums_in_cod1 g).(n,m+1)
        +(Partial_Sums_in_cod1(Partial_Sums_in_cod2 g)).(n,m) ) by Th47
   .= -(Partial_Sums_in_cod1 g).(n,m+1)
        -(Partial_Sums_in_cod1(Partial_Sums_in_cod2 g)).(n,m) by XXREAL_3:25
   .= (-(Partial_Sums_in_cod1 g)).(n,m+1)
        -(Partial_Sums_in_cod1(Partial_Sums_in_cod2 g)).(n,m)
         by A4,A3,MESFUNC1:def 7
   .= (-(Partial_Sums_in_cod1 g)).(n,m+1)
        + (-(Partial_Sums_in_cod1(Partial_Sums_in_cod2 g))).(n,m)
         by A3,A2,MESFUNC1:def 7
   .= (Partial_Sums_in_cod1(-g)).(n,m+1)
        + (-(Partial_Sums_in_cod1(Partial_Sums_in_cod2 g))).(n,m) by Th42
   .= (Partial_Sums_in_cod1 f).(n,m+1)
        + (-(Partial_Sums_in_cod1(Partial_Sums_in_cod2 g))).(n,m) by Th2
   .= (Partial_Sums_in_cod1 f).(n,m+1)
        + ((Partial_Sums_in_cod1-(Partial_Sums_in_cod2 g))).(n,m) by Th42
   .= (Partial_Sums_in_cod1 f).(n,m+1)
        + ((Partial_Sums_in_cod1(Partial_Sums_in_cod2(-g)))).(n,m) by Th42
   .= (Partial_Sums_in_cod1 f).(n,m+1)
        + ((Partial_Sums_in_cod1(Partial_Sums_in_cod2 f))).(n,m) by Th2;
end;
