
theorem Th3:
for f1 be without-infty ExtREAL_sequence, f2 be without+infty ExtREAL_sequence
 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 ExtREAL_sequence,
        f2 be without+infty ExtREAL_sequence;
    set P1 = Partial_Sums f1;
    set P2 = Partial_Sums f2;
    set P12 = Partial_Sums(f1-f2);
    set P21 = Partial_Sums(f2-f1);

    defpred C[Nat] means P12.$1 = P1.$1 - P2.$1;
    P12.0 = (f1-f2).0 by MESFUNC9:def 1; then
    P12.0 = f1.0 - f2.0 by DBLSEQ_3:7; then
    P12.0 = P1.0 - f2.0 by MESFUNC9:def 1; then
A1: C[0] by MESFUNC9:def 1;

A2: for k be Nat st C[k] holds C[k+1]
    proof
     let k be Nat;
     assume A3: C[k];
A4:  P1.k <> -infty & P2.k <> +infty & f1.(k+1) <> -infty
   & f2.(k+1) <> +infty & (f1-f2).(k+1) <> -infty & P2.(k+1) <> +infty
       by MESFUNC5:def 5,def 6; then
A5:  P2.k + f2.(k+1) <> +infty & -f1.(k+1) <> +infty
   & -P2.(k+1) <> -infty by XXREAL_3:16,23;
     P12.(k+1) = P12.k + (f1-f2).(k+1) by MESFUNC9:def 1
      .= P1.k - ( P2.k - (f1-f2).(k+1) ) by A3,A4,XXREAL_3:32
      .= P1.k - ( P2.k - (f1.(k+1) - f2.(k+1)) ) by DBLSEQ_3:7
      .= P1.k - ( P2.k - f1.(k+1) + f2.(k+1) ) by A4,XXREAL_3:32
      .= P1.k - ( - f1.(k+1) + P2.k  + f2.(k+1) ) by XXREAL_3:def 4
      .= P1.k - ( - f1.(k+1) + (P2.k + f2.(k+1)) ) by A4,A5,XXREAL_3:29
      .= P1.k - ( - f1.(k+1) + P2.(k+1) ) by MESFUNC9:def 1
      .= P1.k - ( P2.(k+1) - f1.(k+1) ) by XXREAL_3:def 4
      .= P1.k - P2.(k+1) + f1.(k+1) by A4,XXREAL_3:32
      .= P1.k + (- P2.(k+1)) + f1.(k+1) by XXREAL_3:def 4
      .= P1.k + f1.(k+1) + (- P2.(k+1)) by A4,A5,XXREAL_3:29
      .= P1.(k+1) + (- P2.(k+1)) by MESFUNC9:def 1;
     hence C[k+1] by XXREAL_3:def 4;
    end;
A6: for k be Nat holds C[k] from NAT_1:sch 2(A1,A2);
    for k be Element of NAT holds P12.k = (P1-P2).k
    proof
     let k be Element of NAT;
     P12.k = P1.k - P2.k by A6;
     hence P12.k = (P1-P2).k by DBLSEQ_3:7;
    end;
    hence P12 = P1-P2 by FUNCT_2:63;

    defpred C[Nat] means P21.$1 = P2.$1 - P1.$1;
    P21.0 = (f2-f1).0 by MESFUNC9:def 1
     .= f2.0 - f1.0 by DBLSEQ_3:7
     .= P2.0 - f1.0 by MESFUNC9:def 1; then
A7: C[0] by MESFUNC9:def 1;
A8: for k be Nat st C[k] holds C[k+1]
    proof
     let k be Nat;
     assume A9: C[k];
A10:  P2.k <> +infty & P1.k <> -infty & f2.(k+1) <> +infty
   & f1.(k+1) <> -infty & (f2-f1).(k+1) <> +infty & P1.(k+1) <> -infty
         by MESFUNC5:def 5,def 6; then
A11:  P1.k + f1.(k+1) <> -infty & -f2.(k+1) <> -infty
   & -P1.(k+1) <> +infty by XXREAL_3:17,23;
     P21.(k+1) = P21.k + (f2-f1).(k+1) by MESFUNC9:def 1
      .= P2.k - ( P1.k - (f2-f1).(k+1) ) by A9,A10,XXREAL_3:32
      .= P2.k - ( P1.k - (f2.(k+1) - f1.(k+1)) ) by DBLSEQ_3:7
      .= P2.k - ( P1.k - f2.(k+1) + f1.(k+1) ) by A10,XXREAL_3:32
      .= P2.k - ( - f2.(k+1) + P1.k  + f1.(k+1) ) by XXREAL_3:def 4
      .= P2.k - ( - f2.(k+1) + (P1.k + f1.(k+1)) ) by A10,A11,XXREAL_3:29
      .= P2.k - ( - f2.(k+1) + P1.(k+1) ) by MESFUNC9:def 1
      .= P2.k - ( P1.(k+1) - f2.(k+1) ) by XXREAL_3:def 4
      .= P2.k - P1.(k+1) + f2.(k+1) by A10,XXREAL_3:32
      .= P2.k + (- P1.(k+1)) + f2.(k+1) by XXREAL_3:def 4
      .= P2.k + f2.(k+1) + (- P1.(k+1)) by A10,A11,XXREAL_3:29
      .= P2.(k+1) + (- P1.(k+1)) by MESFUNC9:def 1;
     hence C[k+1] by XXREAL_3:def 4;
    end;
A12: for k be Nat holds C[k] from NAT_1:sch 2(A7,A8);
    for k be Element of NAT holds P21.k = (P2-P1).k
    proof
     let k be Element of NAT;
     P21.k = P2.k - P1.k by A12;
     hence P21.k = (P2-P1).k by DBLSEQ_3:7;
    end;
    hence P21 = P2-P1 by FUNCT_2:63;
end;
