
theorem Th42:
  for f be Function of [:NAT,NAT:],ExtREAL holds
    Partial_Sums_in_cod2(-f) = -(Partial_Sums_in_cod2 f)
  & Partial_Sums_in_cod1(-f) = -(Partial_Sums_in_cod1 f)
proof
   let f be Function of [:NAT,NAT:],ExtREAL;
A1:dom(-(Partial_Sums_in_cod2 f)) = [:NAT,NAT:]
 & dom(-(Partial_Sums_in_cod1 f)) = [:NAT,NAT:] by FUNCT_2:def 1;
A2:dom(-f) = [:NAT,NAT:] by FUNCT_2:def 1;
   for z be Element of [:NAT,NAT:] holds
    (-(Partial_Sums_in_cod2 f)).z = (Partial_Sums_in_cod2(-f)).z
   proof
    let z be Element of [:NAT,NAT:];
    consider n,m be object such that
A3:  n in NAT & m in NAT & z = [n,m] by ZFMISC_1:def 2;
    reconsider n,m as Element of NAT by A3;
    defpred P[Nat] means (Partial_Sums_in_cod2(-f)).(n,$1)
      = -((Partial_Sums_in_cod2 f).(n,$1));
    reconsider z0 = [n,0] as Element of [:NAT,NAT:] by ZFMISC_1:87;
A4: [n,0] in [:NAT,NAT:] by ZFMISC_1:87;
    (Partial_Sums_in_cod2(-f)).(n,0)
      = (-f).(n,0) by DefCSM
     .= -(f.(n,0)) by A4,A2,MESFUNC1:def 7; then
A5: P[0] by DefCSM;
A6: for k be Nat st P[k] holds P[k+1]
    proof
     let k be Nat;
     assume A7: P[k];
A8:  [n,k+1] in [:NAT,NAT:] by ZFMISC_1:87;
     thus (Partial_Sums_in_cod2 -f).(n,k+1)
      = (Partial_Sums_in_cod2 -f).(n,k) + (-f).(n,k+1) by DefCSM
     .= -((Partial_Sums_in_cod2 f).(n,k)) -(f.(n,k+1))
        by A7,A8,A2,MESFUNC1:def 7
     .= -((Partial_Sums_in_cod2 f).(n,k) + f.(n,k+1)) by XXREAL_3:25
     .= -( (Partial_Sums_in_cod2 f).(n,k+1) ) by DefCSM;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2(A5,A6); then
    (Partial_Sums_in_cod2(-f)).(n,m) = -((Partial_Sums_in_cod2 f).(n,m));
    hence thesis by A3,A1,MESFUNC1:def 7;
   end;
   hence Partial_Sums_in_cod2(-f) = -(Partial_Sums_in_cod2 f) by FUNCT_2:def 8;
   for z be Element of [:NAT,NAT:] holds
    (-(Partial_Sums_in_cod1 f)).z = (Partial_Sums_in_cod1(-f)).z
   proof
    let z be Element of [:NAT,NAT:];
    consider n,m be object such that
A3:  n in NAT & m in NAT & z = [n,m] by ZFMISC_1:def 2;
    reconsider n,m as Element of NAT by A3;
    defpred P[Nat] means (Partial_Sums_in_cod1(-f)).($1,m)
      = -((Partial_Sums_in_cod1 f).($1,m));
    reconsider z0 = [0,m] as Element of [:NAT,NAT:] by ZFMISC_1:87;
A4: [0,m] in [:NAT,NAT:] by ZFMISC_1:87;
    (Partial_Sums_in_cod1(-f)).(0,m)
      = (-f).(0,m) by DefRSM
     .= -(f.(0,m)) by A4,A2,MESFUNC1:def 7; then
A5: P[0] by DefRSM;
A6: for k be Nat st P[k] holds P[k+1]
    proof
     let k be Nat;
     assume A7: P[k];
A8:  [k+1,m] in [:NAT,NAT:] by ZFMISC_1:87;
     thus (Partial_Sums_in_cod1 -f).(k+1,m)
      = (Partial_Sums_in_cod1 -f).(k,m) + (-f).(k+1,m) by DefRSM
     .= -((Partial_Sums_in_cod1 f).(k,m)) -(f.(k+1,m))
       by A7,A8,A2,MESFUNC1:def 7
     .= -((Partial_Sums_in_cod1 f).(k,m) + f.(k+1,m)) by XXREAL_3:25
     .= -( (Partial_Sums_in_cod1 f).(k+1,m) ) by DefRSM;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2(A5,A6); then
    (Partial_Sums_in_cod1(-f)).(n,m) = -((Partial_Sums_in_cod1 f).(n,m));
    hence thesis by A3,A1,MESFUNC1:def 7;
   end;
   hence Partial_Sums_in_cod1(-f) = -(Partial_Sums_in_cod1 f) by FUNCT_2:def 8;
end;
