
theorem Th61:
  for J,K,L be ExtREAL_sequence st (for n,m be Nat st n <=m holds
J.n <= J.m) & (for n,m be Nat st n <=m holds K.n <= K.m) & J is without-infty &
K is without-infty & (for n be Nat holds J.n+K.n =L.n ) holds L is convergent &
  lim L= sup rng L & lim L = lim J + lim K & sup rng L = sup rng K + sup rng J
proof
  let J,K,L be ExtREAL_sequence;
  assume that
A1: for n,m be Nat st n <=m holds J.n <= J.m and
A2: for n,m be Nat st n <=m holds K.n <= K.m and
A3: J is without-infty and
A4: K is without-infty and
A5: for n be Nat holds J.n+K.n =L.n;
A6: dom K = NAT by FUNCT_2:def 1;
A7: dom J = NAT by FUNCT_2:def 1;
A8: now
    per cases by A3,Lm8;
    suppose
A9:   sup rng J in REAL;
      then reconsider SJ = sup rng J as Real;
      per cases by A4,Lm8;
      suppose
A10:    sup rng K in REAL;
        then reconsider SK = sup rng K as Real;
A11:    now
          let p be Real;
          assume
A12:      0 < p;
          then consider SJ9 be ExtReal such that
A13:      SJ9 in rng J and
A14:      sup rng J - p/2 < SJ9 by A9,MEASURE6:6;
          consider nj be object such that
A15:      nj in dom J and
A16:      SJ9 = J.nj by A13,FUNCT_1:def 3;
          reconsider nj as Element of NAT by A15;
          consider SK9 be ExtReal such that
A17:      SK9 in rng K and
A18:      sup rng K - p/2 < SK9 by A10,A12,MEASURE6:6;
          consider nk be object such that
A19:      nk in dom K and
A20:      SK9 = K.nk by A17,FUNCT_1:def 3;
          reconsider nk as Element of NAT by A19;
          reconsider n = max(nj,nk) as Nat;
          take n;
          hereby
            reconsider SJ9, SK9 as R_eal by XXREAL_0:def 1;
            let m be Nat;
            assume
A21:        n <= m;
            nk <=n by XXREAL_0:25;
            then nk <= m by A21,XXREAL_0:2;
            then SK9 <= K.m by A2,A20;
            then
A22:        SK - K.m <= SK - SK9 by XXREAL_3:37;
            nj <= n by XXREAL_0:25;
            then nj <= m by A21,XXREAL_0:2;
            then SJ9 <= J.m by A1,A16;
            then SJ - J.m <= SJ - SJ9 by XXREAL_3:37;
            then
A23:        (SJ - J.m) + (SK - K.m) <= (SJ - SJ9) + (
            SK - SK9) by A22,XXREAL_3:36;
            SJ in REAL by XREAL_0:def 1;
            then
A24:         SJ < +infty by XXREAL_0:9;
            reconsider s1 = SK as Element of REAL by XREAL_0:def 1;
            reconsider m1=m as Element of NAT by ORDINAL1:def 12;
A25:        -(L.m - (SJ+SK)) = (SJ+SK) - L.m by XXREAL_3:26;
A26:          (p/2) in REAL by XREAL_0:def 1;
             SK < (p/2) + SK9 by A18,XXREAL_3:54;
            then SK - SK9 < (p/2) by XXREAL_3:55;
            then
A27:        (p/2) + (SK - SK9) < (p/2) + (p/2) by
XXREAL_3:43,A26;
             SJ < (p/2) + SJ9 by A14,XXREAL_3:54;
            then
A28:        SJ - SJ9 < (p/2) by XXREAL_3:55;
            nk <= n by XXREAL_0:25;
            then nk <= m by A21,XXREAL_0:2;
            then
A29:        K.nk <= K.m by A2;
A30:        SK in REAL by XREAL_0:def 1;
            then
A31:        SK < +infty by XXREAL_0:9;
            K.m1 in rng K by A6,FUNCT_1:3;
            then
A32:        K.m <= SK by XXREAL_2:4;
            then
A33:        K.m < +infty by A30,XXREAL_0:2,9;
            -infty < SK9 by A4,A20;
            then reconsider s0 = SK9 as Element of REAL
               by A20,A33,A29,XXREAL_0:14;
A34:        L.m = J.m + K.m by A5;
            J.m1 in rng J by A7,FUNCT_1:3;
            then
A35:        J.m <= SJ by XXREAL_2:4;
            then J.m + K.m <= SJ + SK by A32,XXREAL_3:36;
            then L.m - (SJ+SK) <= 0 by A34,A25,XXREAL_3:40;
            then
A36:        |. L.m - (SJ+SK) .| = (SJ+SK) - L.m by A25,EXTREAL1:18;
            SK - SK9 = s1 - s0 by SUPINF_2:3;
            then (SJ - SJ9) + (SK - SK9) < (p/2) + (SK
            - SK9) by A28,XXREAL_3:43;
            then
A37:        (SJ - SJ9) + (SK - SK9) < (p/2) + (p/2)
            by A27,XXREAL_0:2;
            -infty < K.m by A4;
            then
            SJ - J.m + (SK - K.m) = SJ - J.m + SK
            - K.m by A33,XXREAL_3:30
              .= SK + SJ - J.m - K.m by XXREAL_3:30
              .= (SK+SJ) - (J.m + K.m) by A24,A31,A35,A32,XXREAL_3:31
              .= (SK+SJ) - L.m by A5;
            hence |. L.m - (SJ+SK) .| < p by A36,A37,A23,XXREAL_0:2;
          end;
        end;
        then
A38:    L is convergent_to_finite_number;
        hence L is convergent;
        hence lim L = sup rng J + sup rng K by A11,A38,Def12;
      end;
      suppose
A39:    sup rng K = +infty;
        for p be Real st 0 < p ex n be Nat st for m be Nat st n<=m
        holds p <= L.m
        proof
          reconsider supj = sup rng J as Element of REAL by A9;
          let p be Real;
          reconsider p92 = p/2, p9 = p as Element of REAL by XREAL_0:def 1;
          assume 0 < p;
          then consider j be ExtReal such that
A40:      j in rng J and
A41:      sup rng J - (p/2) < j by A9,MEASURE6:6;
          consider n1 being object such that
A42:      n1 in dom J and
A43:      j = J.n1 by A40,FUNCT_1:def 3;
A44:      supj - p92 = sup rng J - (p/2) by SUPINF_2:3;
          then
A45:      p9 - (supj - p92) = p - (sup rng J - (p/2)) by SUPINF_2:3;
          then p - (sup rng J -(p/2)) < sup rng K by A39,XXREAL_0:9;
          then consider k being Element of ExtREAL such that
A46:      k in rng K and
A47:      p - (sup rng J - (p/2)) < k by XXREAL_2:94;
          p9 = (p9 - (supj - p92)) + (supj - p92);
          then
A48:      p - (sup rng J - (p/2)) + (sup rng J - (p/2)) =
          p9 by A44,A45,SUPINF_2:1;
          reconsider n1 as Element of NAT by A42;
          consider n2 being object such that
A49:      n2 in dom K and
A50:      k = K.n2 by A46,FUNCT_1:def 3;
          reconsider n2 as Element of NAT by A49;
          set n = max(n1,n2);
          J.n1 <= J.n by A1,XXREAL_0:25;
          then
A51:      sup rng J - (p/2) < J.n by A41,A43,XXREAL_0:2;
          K.n2 <= K.n by A2,XXREAL_0:25;
          then
A52:      p - (sup rng J - (p/2)) < K.n by A47,A50,XXREAL_0:2;
A53:      p < J.n + K.n by A51,A52,A48,XXREAL_3:64;
          now
            let m be Nat;
            assume
A54:        n <= m;
            then
A55:        K.n <= K.m by A2;
            J.n <= J.m by A1,A54;
            then J.n + K.n <= J.m + K.m by A55,XXREAL_3:36;
            then J.n + K.n <= L.m by A5;
            hence p <= L.m by A53,XXREAL_0:2;
          end;
          hence thesis;
        end;
        then
A56:    L is convergent_to_+infty;
        hence L is convergent;
        then lim L = +infty by A56,Def12;
        hence lim L = sup rng J + sup rng K by A9,A39,XXREAL_3:def 2;
      end;
    end;
    suppose
A57:  sup rng J = +infty;
      now
        let p be Real;
        assume
A58:    0 < p;
        per cases by A4,Lm8;
        suppose
A59:      sup rng K in REAL;
          then reconsider supk = sup rng K as Element of REAL;
          reconsider p92 = p/2, p9 = p as Element of REAL by XREAL_0:def 1;
A60:      supk - p92 = sup rng K - (p/2) by SUPINF_2:3;
          then
A61:      p9 - (supk - p92) = p - (sup rng K - (p/2)) by SUPINF_2:3;
          then p - (sup rng K - (p/2)) < sup rng J by A57,XXREAL_0:9;
          then consider j being Element of ExtREAL such that
A62:      j in rng J and
A63:      p - (sup rng K - (p/2)) < j by XXREAL_2:94;
          p9 = p9 - (supk - p92) + (supk - p92);
          then
A64:      p - (sup rng K - (p/2)) + (sup rng K - (p/2))
          = p9 by A60,A61,SUPINF_2:1;
          consider k be ExtReal such that
A65:      k in rng K and
A66:      sup rng K - (p/2) < k by A58,A59,MEASURE6:6;
          consider n1 being object such that
A67:      n1 in dom K and
A68:      k = K.n1 by A65,FUNCT_1:def 3;
          consider n2 being object such that
A69:      n2 in dom J and
A70:      j = J.n2 by A62,FUNCT_1:def 3;
          reconsider n1 as Element of NAT by A67;
          reconsider n2 as Element of NAT by A69;
          set n = max(n1,n2);
          J.n2 <= J.n by A1,XXREAL_0:25;
          then
A71:      p - (sup rng K - (p/2)) < J.n by A63,A70,XXREAL_0:2;
          K.n1 <= K.n by A2,XXREAL_0:25;
          then
A72:      sup rng K - (p/2) < K.n by A66,A68,XXREAL_0:2;
A73:      p < J.n + K.n by A72,A71,A64,XXREAL_3:64;
          now
            let m be Nat;
            assume
A74:        n <= m;
            then
A75:        K.n <= K.m by A2;
            J.n <= J.m by A1,A74;
            then J.n + K.n <= J.m + K.m by A75,XXREAL_3:36;
            then J.n + K.n <= L.m by A5;
            hence p <= L.m by A73,XXREAL_0:2;
          end;
          hence ex n be Nat st for m be Nat st n <= m holds p <= L.m;
        end;
        suppose
          sup rng K = +infty;
          then consider n1 be Nat such that
A76:      (p/2) < K.n1 by A4,A58,Th59;
          consider n2 be Nat such that
A77:      (p/2) < J.n2 by A3,A57,A58,Th59;
          reconsider n1,n2 as Element of NAT by ORDINAL1:def 12;
          set n = max(n1,n2);
          K.n1 <= K.n by A2,XXREAL_0:25;
          then
A78:      (p/2) < K.n by A76,XXREAL_0:2;
          J.n2 <= J.n by A1,XXREAL_0:25;
          then
A79:      (p/2) < J.n by A77,XXREAL_0:2;
          (p/2) + (p/2) < J.n + K.n by A79,A78,XXREAL_3:64;
          then p < J.n + K.n;
          then
A80:       p < L.n by A5;
          now
            let m be Nat;
            assume
A81:        n <= m;
            then
A82:        K.n <= K.m by A2;
            J.n <= J.m by A1,A81;
            then J.n + K.n <= J.m + K.m by A82,XXREAL_3:36;
            then J.n + K.n <= L.m by A5;
            then L.n <= L.m by A5;
            hence p <= L.m by A80,XXREAL_0:2;
          end;
          hence ex n be Nat st for m be Nat st n <= m holds p <= L.m;
        end;
      end;
      then
A83:  L is convergent_to_+infty;
      hence L is convergent;
      then
A84:  lim L = +infty by A83,Def12;
A85:  K.0 <= sup rng K by A6,FUNCT_1:3,XXREAL_2:4;
      -infty < K.0 by A4;
      hence lim L = sup rng J + sup rng K by A57,A84,A85,XXREAL_3:def 2;
    end;
  end;
  hence L is convergent;
A86: now
    let n,m be Nat;
    assume
A87: n <= m;
    then
A88: K.n <= K.m by A2;
    J.n <= J.m by A1,A87;
    then J.n + K.n <= J.m + K.m by A88,XXREAL_3:36;
    then L.n <= J.m + K.m by A5;
    hence L.n <= L.m by A5;
  end;
  hence lim L= sup rng L by Th54;
  lim J = sup rng J by A1,Th54;
  hence thesis by A2,A8,A86,Th54;
end;
