reserve X for RealUnitarySpace;
reserve x, y, y1, y2 for Point of X;

theorem Th1:
  for Y be Subset of X for L be Functional of X holds Y
is_summable_set_by L iff for e be Real st 0 < e holds ex Y0 be finite Subset of
  X st Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X st Y1 is non
empty & Y1 c= Y & Y0 misses Y1 holds |.setopfunc(Y1, the carrier of X, REAL,
  L, addreal).| <e
proof
  let Y be Subset of X;
  let L be Functional of X;
  thus Y is_summable_set_by L implies for e be Real st 0 < e holds ex Y0 be
finite Subset of X st Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X
st Y1 is non empty & Y1 c= Y & Y0 misses Y1 holds |.setopfunc(Y1, the carrier
  of X, REAL, L, addreal).| <e
  proof
    assume Y is_summable_set_by L;
    then consider r be Real such that
A1: for e be Real
     st e > 0 ex Y0 be finite Subset of X st Y0 is non
empty & Y0 c= Y & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y holds |.
    r - setopfunc(Y1, the carrier of X, REAL, L, addreal).| < e
by BHSP_6:def 6;
    for e be Real st 0 < e
    ex Y0 be finite Subset of X st Y0 is non
empty & Y0 c= Y & for Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y &
Y0 misses Y1 holds |.setopfunc(Y1, the carrier of X, REAL, L, addreal).| <e
    proof
      let e be Real;
      assume 0 < e;
      then consider Y0 be finite Subset of X such that
A2:   Y0 is non empty and
A3:   Y0 c= Y and
A4:   for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y holds |.r -
setopfunc(Y1, the carrier of X, REAL, L, addreal).| < e/2 by A1,XREAL_1:139;
      for Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y & Y0
misses Y1 holds |.setopfunc(Y1, the carrier of X, REAL, L, addreal).| < e
      proof
        let Y1 be finite Subset of X such that
        Y1 is non empty and
A5:     Y1 c= Y and
A6:     Y0 misses Y1;
        set Y19 = Y0 \/ Y1;
        dom L = the carrier of X by FUNCT_2:def 1;
        then setopfunc(Y19, the carrier of X, REAL, L, addreal) = addreal.(
setopfunc(Y0, the carrier of X, REAL, L, addreal), setopfunc(Y1, the carrier of
        X, REAL, L, addreal)) by A6,BHSP_5:14
          .= setopfunc(Y0, the carrier of X, REAL, L, addreal) + setopfunc(
        Y1, the carrier of X, REAL, L, addreal) by BINOP_2:def 9;
        then
A7:     setopfunc(Y1, the carrier of X, REAL, L, addreal) = setopfunc(Y19
, the carrier of X, REAL, L, addreal) - setopfunc(Y0, the carrier of X, REAL, L
        , addreal);
        Y0 c= Y19 by XBOOLE_1:7;
        then
        |.r - setopfunc(Y19, the carrier of X, REAL, L, addreal).| < e/2
        by A3,A4,A5,XBOOLE_1:8;
        then
A8:     |.setopfunc(Y19, the carrier of X, REAL, L, addreal)-r.| < e/2
        by UNIFORM1:11;
        |.r - setopfunc(Y0, the carrier of X, REAL, L, addreal).| < e/2
        by A3,A4;
        hence thesis by A8,A7,Lm1;
      end;
      hence thesis by A2,A3;
    end;
    hence thesis;
  end;
  assume
A9: for e be Real st 0 < e holds ex Y0 be finite Subset of X st Y0 is
non empty & Y0 c= Y & for Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y
& Y0 misses Y1 holds |.setopfunc(Y1, the carrier of X, REAL, L, addreal).| <e;
  ex r be Real st
   for e be Real st 0 < e ex Y0 be finite Subset of X st
Y0 is non empty & Y0 c= Y & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y
  holds |.r - setopfunc(Y1, the carrier of X, REAL, L, addreal).| < e
  proof
    defpred P[object,object] means
    ex D2 being set st D2 = $2 &
    $2 is finite Subset of X & D2 is non empty & D2
c= Y & for z be Real st z=$1 for Y1 be finite Subset of X st Y1 is non empty &
    Y1 c= Y & D2 misses Y1 holds |.setopfunc(Y1, the carrier of X, REAL, L,
    addreal).| < 1/(z+1);
A10: for x be object st x in NAT ex y be object st y in bool Y & P[x,y]
    proof
      let x be object;
      assume x in NAT;
      then reconsider xx = x as Element of NAT;
      reconsider e = 1/(xx+1) as Real;
      0/(xx + 1) < 1/(xx + 1) by XREAL_1:74;
      then consider Y0 be finite Subset of X such that
A11:  Y0 is non empty and
A12:  Y0 c= Y & for Y1 be finite Subset of X st Y1 is non empty & Y1
c= Y & Y0 misses Y1 holds |.setopfunc(Y1, the carrier of X, REAL, L, addreal)
      .| < e by A9;
      take Y0;
      thus Y0 in bool Y by A12,ZFMISC_1:def 1;
      take D2 = Y0;
      thus D2 = Y0;
      thus Y0 is finite Subset of X;
      thus D2  is non empty by A11;
      thus D2 c= Y by A12;
for z be Real st z = x for Y1 be finite Subset of X
st Y1 is non empty & Y1 c= Y & Y0 misses Y1 holds |.setopfunc(Y1, the carrier
      of X, REAL, L, addreal).| < 1/(z+1) by A12;
      hence
for z be Real st z=x
for Y1 be finite Subset of X st Y1 is non
empty & Y1 c= Y & D2 misses Y1 holds |.setopfunc(Y1, the carrier of X, REAL, L
      , addreal).| < 1/(z+1);
    end;
A13: ex B being sequence of bool Y st
for x be object st x in NAT holds P[x,B. x] from FUNCT_2:sch 1(A10);
    ex A be sequence of  bool Y st for i be Nat holds A.i
is finite Subset of X & A.i is non empty & A.i c= Y & (for Y1 be finite Subset
of X st Y1 is non empty & Y1 c= Y & A.i misses Y1 holds |.setopfunc(Y1, the
carrier of X, REAL, L, addreal).| < 1/(i+1)) & for j be Element of NAT
       st i <= j
    holds A.i c= A.j
    proof
      consider B being sequence of bool Y such that
A14:  for x be object st x in NAT holds P[x,B. x] by A13;
A15:  for x be object st x in NAT
    holds B.x is finite Subset of X & B.x
is non empty & B.x c= Y & for z be Real st z=x for Y1 be finite Subset of X st
Y1 is non empty & Y1 c= Y & B.x misses Y1 holds |.setopfunc(Y1, the carrier
      of X, REAL, L, addreal).| < 1/(z+1)
     proof let x be object such that
A16:    x in NAT;
     P[x,B.x] by A16,A14;
     hence B.x is finite Subset of X &
      B.x is non empty & B.x c= Y &
   for z be Real st z=x for Y1 be finite Subset of X st
Y1 is non empty & Y1 c= Y & B.x misses Y1 holds |.setopfunc(Y1, the carrier
      of X, REAL, L, addreal).| < 1/(z+1);
     end;
      deffunc G(Nat,set) = B.($1+1) \/ $2;
      ex A being Function st dom A = NAT & A.0 = B.0 & for n being Nat
      holds A.(n+1) = G(n,A.n) from NAT_1:sch 11;
      then consider A being Function such that
A17:  dom A = NAT and
A18:  A.0 = B.0 and
A19:  for n being Nat holds A.(n+1) = B.(n+1) \/ A.n;
      defpred R[Nat] means
A.$1 is finite Subset of X & A.$1 is non
empty & A.$1 c= Y & (for Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y
& A.$1 misses Y1 holds |.setopfunc(Y1, the carrier of X, REAL, L, addreal).| <
      1/($1+1)) & for j be Element of NAT st $1 <= j holds A.$1 c= A.j;
A20:  now
        let n be Nat such that
A21:    R[n];
A22:    for Y1 be finite Subset of X st Y1 is non empty & Y1 c= Y & A.(n+
1) misses Y1 holds |.setopfunc(Y1, the carrier of X, REAL, L, addreal).| < 1/(
        (n+1)+1)
        proof
          let Y1 be finite Subset of X such that
A23:      Y1 is non empty & Y1 c= Y and
A24:      A.(n+1) misses Y1;
          A.(n+1) = B.(n+1) \/ A.n by A19;
          then B.(n+1) misses Y1 by A24,XBOOLE_1:7,63;
          hence thesis by A15,A23;
        end;
        defpred P[Nat] means n+1 <= $1 implies A.(n+1) c= A.$1;
A25:    for j being Nat st P[j] holds P[j+1]
        proof
          let j be Nat such that
A26:      P[j] and
A27:      n+1 <= j+1;
          now
            per cases;
            case
              n = j;
              hence thesis;
            end;
            case
A28:          n <> j;
              A.(j+1) = (B.(j+1) \/ A.j) by A19;
              then
A29:          A.j c= A.(j+1) by XBOOLE_1:7;
              n <= j by A27,XREAL_1:6;
              then n < j by A28,XXREAL_0:1;
              hence thesis by A26,A29,NAT_1:13;
            end;
          end;
          hence thesis;
        end;
A30:    P[0];
A31:    for j be Nat holds P[j] from NAT_1:sch 2(A30, A25);
        A.(n+1) = B.(n+1) \/ A.n & B.(n+1) is finite Subset of X by A15,A19;
        hence R[n+1] by A21,A22,A31,XBOOLE_1:8;
      end;
      for j0 be Nat st j0=0 holds for j be Nat st j0 <= j holds A.j0 c= A.j
      proof
        let j0 be Nat such that
A32:    j0 = 0;
        defpred P[Nat] means j0 <= $1 implies A.j0 c= A.$1;
A33:    now
          let j be Nat such that
A34:      P[j];
          A.(j+1) = (B.(j+1) \/ A.j) by A19;
          then A.j c= A.(j+1) by XBOOLE_1:7;
          hence P[j+1] by A32,A34,XBOOLE_1:1;
        end;
A35:    P[0] by A32;
        thus for j be Nat holds P[j] from NAT_1:sch 2(A35, A33 );
      end;
      then
A36:  R[0] by A15,A18;
A37:  for i be Nat holds R[i] from NAT_1:sch 2(A36,A20);
      rng A c= bool Y
      proof
        let y be object;
        assume y in rng A;
        then consider x be object such that
A38:    x in dom A and
A39:    y = A.x by FUNCT_1:def 3;
        reconsider i = x as Element of NAT by A17,A38;
        A.i c= Y by A37;
        hence thesis by A39,ZFMISC_1:def 1;
      end;
      then A is sequence of  bool Y by A17,FUNCT_2:2;
      hence thesis by A37;
    end;
    then consider A be sequence of  bool Y such that
A40: for i be Nat holds A.i is finite Subset of X & A.i is
non empty & A.i c= Y & (for Y1 be finite Subset of X st Y1 is non empty & Y1 c=
Y & A.i misses Y1 holds |.setopfunc(Y1, the carrier of X, REAL, L, addreal).|<
    1/(i+1)) & for j be Element of NAT st i <= j holds A.i c= A.j;
    defpred P[object,object] means
ex Y1 be finite Subset of X st Y1 is non empty &
    A.$1 = Y1 & $2 = setopfunc(Y1, the carrier of X, REAL, L, addreal);
A41: for x be object st x in NAT ex y be object st y in REAL & P[x,y]
    proof
      let x be object;
      assume x in NAT;
      then reconsider i=x as Element of NAT;
      A.i is finite Subset of X by A40;
      then reconsider Y1 = A.x as finite Subset of X;
      reconsider y = setopfunc(Y1, the carrier of X, REAL, L, addreal) as set;
      A.i is non empty by A40;
      then ex Y1 be finite Subset of X st Y1 is non empty & A.x = Y1 & y =
      setopfunc(Y1, the carrier of X, REAL, L, addreal);
      hence thesis;
    end;
    ex F being sequence of  REAL st for x be object st x in NAT
      holds P[x,F.x] from FUNCT_2:sch 1(A41);
    then consider F being sequence of  REAL such that
A42: for x be object st x in NAT holds ex Y1 be finite Subset of X st Y1
    is non empty & A.x = Y1 & F.x = setopfunc(Y1, the carrier of X, REAL, L,
    addreal);
    set seq = F;
A43: for e be Real st e > 0 ex k be Nat st
   for n, m be Nat st n >= k & m >= k
      holds |.(seq.n) - ((seq.m) qua Real).| < e
    proof
      let e be Real such that
A44:  e > 0;
A45:  e/2 > 0/2 by A44,XREAL_1:74;
      then consider k be Nat such that
A46:  1/(k+1) < e/2 by Lm2;
      take k;
      let nn, mm be Nat such that
A47:  nn >= k and
A48:  mm >= k;
      reconsider m=mm, n=nn, k as Element of NAT by ORDINAL1:def 12;
      consider Y2 be finite Subset of X such that
      Y2 is non empty and
A49:  A.m = Y2 and
A50:  seq.m = setopfunc(Y2, the carrier of X, REAL, L, addreal) by A42;
      consider Y0 be finite Subset of X such that
      Y0 is non empty and
A51:  A.k = Y0 and
      seq.k = setopfunc(Y0, the carrier of X, REAL, L, addreal) by A42;
A52:  Y0 c= Y2 by A40,A48,A51,A49;
      consider Y1 be finite Subset of X such that
      Y1 is non empty and
A53:  A.n = Y1 and
A54:  seq.n = setopfunc(Y1, the carrier of X, REAL, L, addreal) by A42;
A55:  Y0 c= Y1 by A40,A47,A51,A53;
      now
        per cases;
        case
A56:      Y0 = Y1;
          now
            per cases;
            case
              Y0 = Y2;
              hence thesis by A44,A54,A50,A56,ABSVALUE:2;
            end;
            case
A57:          Y0 <> Y2;
              ex Y02 be finite Subset of X st Y02 is non empty & Y02 c= Y
              & Y02 misses Y0 & Y0 \/ Y02 = Y2
              proof
                take Y02 = Y2 \ Y0;
A58:            Y2 \ Y0 c= Y2 by XBOOLE_1:36;
                Y0 \/ Y02 = Y0 \/ Y2 by XBOOLE_1:39
                  .= Y2 by A52,XBOOLE_1:12;
                hence thesis by A49,A57,A58,XBOOLE_1:1,79;
              end;
              then consider Y02 be finite Subset of X such that
A59:          Y02 is non empty & Y02 c= Y and
A60:          Y02 misses Y0 and
A61:          Y0 \/ Y02 = Y2;
              |.setopfunc(Y02, the carrier of X, REAL, L, addreal).| <
              1/(k+1) by A40,A51,A59,A60;
              then
A62:          |.setopfunc(Y02, the carrier of X, REAL, L, addreal).| <
              e/2 by A46,XXREAL_0:2;
              dom L = the carrier of X by FUNCT_2:def 1;
              then setopfunc(Y2, the carrier of X, REAL, L, addreal) =
addreal.( setopfunc(Y0, the carrier of X, REAL, L, addreal), setopfunc(Y02, the
              carrier of X, REAL, L, addreal)) by A60,A61,BHSP_5:14
                .= setopfunc(Y0, the carrier of X, REAL, L, addreal) +
setopfunc(Y02, the carrier of X, REAL, L, addreal) by BINOP_2:def 9;
              then
A63:          |.(seq.n) - (seq.m).| = |.-setopfunc(Y02, the carrier
              of X, REAL, L, addreal).| by A54,A50,A56
                .= |.setopfunc(Y02, the carrier of X, REAL, L, addreal).|
              by COMPLEX1:52;
              e/2 < e by A44,XREAL_1:216;
              hence thesis by A62,A63,XXREAL_0:2;
            end;
          end;
          hence thesis;
        end;
        case
A64:      Y0 <> Y1;
          now
            per cases;
            case
A65:          Y0 = Y2;
              ex Y01 be finite Subset of X st Y01 is non empty & Y01 c=
              Y & Y01 misses Y0 & Y0 \/ Y01 = Y1
              proof
                take Y01 = Y1 \ Y0;
A66:            Y1 \ Y0 c= Y1 by XBOOLE_1:36;
                Y0 \/ Y01 = Y0 \/ Y1 by XBOOLE_1:39
                  .= Y1 by A55,XBOOLE_1:12;
                hence thesis by A53,A64,A66,XBOOLE_1:1,79;
              end;
              then consider Y01 be finite Subset of X such that
A67:          Y01 is non empty & Y01 c= Y and
A68:          Y01 misses Y0 and
A69:          Y0 \/ Y01 = Y1;
              dom L = the carrier of X by FUNCT_2:def 1;
              then
A70:          setopfunc(Y1, the carrier of X, REAL, L, addreal) =
addreal.( setopfunc(Y0, the carrier of X, REAL, L, addreal), setopfunc(Y01, the
              carrier of X, REAL, L, addreal)) by A68,A69,BHSP_5:14
                .= setopfunc(Y0, the carrier of X, REAL, L, addreal) +
setopfunc(Y01, the carrier of X, REAL, L, addreal) by BINOP_2:def 9;
              |.setopfunc(Y01, the carrier of X, REAL, L, addreal).| <
              1/(k+1) by A40,A51,A67,A68;
              then
              |.(seq.n) - (seq.m).| < e/2 by A46,A54,A50,A65,A70,XXREAL_0:2;
              then |.(seq.n) - (seq.m).|+ 0 < e/2 + e/2 by A45,XREAL_1:8;
              hence thesis;
            end;
            case
A71:          Y0<>Y2;
              ex Y02 be finite Subset of X st Y02 is non empty & Y02 c=
              Y & Y02 misses Y0 & Y0 \/ Y02 = Y2
              proof
                take Y02 = Y2 \ Y0;
A72:            Y2 \ Y0 c= Y2 by XBOOLE_1:36;
                Y0 \/ Y02 = Y0 \/ Y2 by XBOOLE_1:39
                  .= Y2 by A52,XBOOLE_1:12;
                hence thesis by A49,A71,A72,XBOOLE_1:1,79;
              end;
              then consider Y02 be finite Subset of X such that
A73:          Y02 is non empty & Y02 c= Y and
A74:          Y02 misses Y0 and
A75:          Y0 \/ Y02 = Y2;
              dom L = the carrier of X by FUNCT_2:def 1;
              then
A76:          setopfunc(Y2, the carrier of X, REAL, L, addreal) =
addreal.(setopfunc(Y0, the carrier of X, REAL, L, addreal), setopfunc(Y02, the
              carrier of X, REAL, L, addreal)) by A74,A75,BHSP_5:14
                .= setopfunc(Y0, the carrier of X, REAL, L, addreal) +
setopfunc(Y02, the carrier of X, REAL, L, addreal) by BINOP_2:def 9;
              ex Y01 be finite Subset of X st Y01 is non empty & Y01 c=
              Y & Y01 misses Y0 & Y0 \/ Y01 = Y1
              proof
                take Y01 = Y1 \ Y0;
A77:            Y1 \ Y0 c= Y1 by XBOOLE_1:36;
                Y0 \/ Y01 = Y0 \/ Y1 by XBOOLE_1:39
                  .= Y1 by A55,XBOOLE_1:12;
                hence thesis by A53,A64,A77,XBOOLE_1:1,79;
              end;
              then consider Y01 be finite Subset of X such that
A78:          Y01 is non empty & Y01 c= Y and
A79:          Y01 misses Y0 and
A80:          Y0 \/ Y01 = Y1;
              dom L = the carrier of X by FUNCT_2:def 1;
              then setopfunc(Y1, the carrier of X, REAL, L, addreal) =
addreal.(setopfunc(Y0, the carrier of X, REAL, L, addreal), setopfunc(Y01, the
              carrier of X, REAL, L, addreal)) by A79,A80,BHSP_5:14
                .= setopfunc(Y0, the carrier of X, REAL, L, addreal) +
setopfunc(Y01, the carrier of X, REAL, L, addreal) by BINOP_2:def 9;
              then seq.n - seq.m = setopfunc(Y01, the carrier of X, REAL, L,
addreal) - setopfunc(Y02, the carrier of X, REAL, L, addreal) by A54,A50,A76;
              then
A81:          |.(seq.n) - (seq.m).| <= |.setopfunc(Y01, the carrier
of X, REAL, L, addreal).| + |.setopfunc(Y02, the carrier of X, REAL, L,
              addreal).| by COMPLEX1:57;
              |.setopfunc(Y02, the carrier of X, REAL, L, addreal).| <
              1/(k+1) by A40,A51,A73,A74;
              then
A82:          |.setopfunc(Y02, the carrier of X, REAL, L, addreal).| <
              e/2 by A46,XXREAL_0:2;
              |.setopfunc(Y01, the carrier of X, REAL, L, addreal).| <
              1/(k+1) by A40,A51,A78,A79;
              then |.setopfunc(Y01, the carrier of X, REAL, L, addreal).| <
              e/2 by A46,XXREAL_0:2;
              then |.setopfunc(Y01, the carrier of X, REAL, L, addreal).| +
|.setopfunc(Y02, the carrier of X, REAL, L, addreal).| < e/2 + e/2 by A82,
XREAL_1:8;
              hence thesis by A81,XXREAL_0:2;
            end;
          end;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
    for e be Real st 0 < e ex k be Nat st
     for m be Nat st k <= m holds |.seq.m -seq.k.|<e
    proof
      let e be Real;
      assume 0 < e;
      then consider k be Nat such that
A83:  for n, m be Nat st n >= k & m >= k holds |.(seq.n) - (seq.m).| < e
                by A43;
      for m be Nat st k <= m holds |.seq.m - seq.k.|<e by A83;
      hence thesis;
    end;
    then seq is convergent by SEQ_4:41;
    then consider x be Real such that
A84: for r be Real st r > 0 ex m be Nat st for n be
    Nat st n >= m holds |.seq.n - x.| < r by SEQ_2:def 6;
    reconsider r=x as Real;
    take r;
    for e be Real st 0 < e ex Y0 be finite Subset of X st Y0 is non
empty & Y0 c= Y & for Y1 be finite Subset of X st Y0 c= Y1 & Y1 c= Y holds |.
    r - setopfunc(Y1, the carrier of X, REAL, L, addreal).| < e
    proof
      let e be Real;
      assume e > 0;
      then
A85:  e/2 > 0/2 by XREAL_1:74;
      then consider m be Nat such that
A86:  for n be Nat st n >= m holds |.(seq.n)-r.| < e/2 by A84;
      consider i be Nat such that
A87:  1/(i+1) < e/2 and
A88:  i >= m by A85,Lm3;
    i in NAT by ORDINAL1:def 12;
      then reconsider ii=i as Element of NAT;
      consider Y0 be finite Subset of X such that
A89:  Y0 is non empty and
A90:  A.ii = Y0 and
A91:  seq.i = setopfunc(Y0, the carrier of X, REAL, L, addreal) by A42;
A92:  |.setopfunc(Y0, the carrier of X, REAL, L, addreal) - r.| < e/2
      by A86,A88,A91;
      now
        let Y1 be finite Subset of X such that
A93:    Y0 c= Y1 and
A94:    Y1 c= Y;
        now
          per cases;
          case
            Y0 = Y1;
            then |.r - setopfunc(Y1, the carrier of X, REAL, L, addreal).| <
            e/2 by A92,UNIFORM1:11;
            then |.x - setopfunc(Y1, the carrier of X, REAL, L, addreal).| +
            0 < e/2 + e/2 by A85,XREAL_1:8;
            hence |.r - setopfunc(Y1, the carrier of X, REAL, L, addreal).| <
            e;
          end;
          case
A95:        Y0 <> Y1;
            ex Y2 be finite Subset of X st Y2 is non empty & Y2 c= Y &
            Y0 misses Y2 & Y0 \/ Y2 = Y1
            proof
              take Y2 = Y1 \ Y0;
A96:          Y1 \ Y0 c= Y1 by XBOOLE_1:36;
              Y0 \/ Y2 = Y0 \/ Y1 by XBOOLE_1:39
                .= Y1 by A93,XBOOLE_1:12;
              hence thesis by A94,A95,A96,XBOOLE_1:79;
            end;
            then consider Y2 be finite Subset of X such that
A97:        Y2 is non empty & Y2 c= Y and
A98:        Y0 misses Y2 and
A99:        Y0 \/ Y2 = Y1;
            dom L = the carrier of X by FUNCT_2:def 1;
            then setopfunc(Y1, the carrier of X, REAL, L, addreal)-r =
addreal.( setopfunc(Y0, the carrier of X, REAL, L, addreal), setopfunc(Y2, the
            carrier of X, REAL, L, addreal)) - r by A98,A99,BHSP_5:14
              .= setopfunc(Y0, the carrier of X, REAL, L, addreal) +
setopfunc(Y2, the carrier of X, REAL, L, addreal) - r by BINOP_2:def 9
              .= setopfunc(Y0, the carrier of X, REAL, L, addreal) - r +
            setopfunc(Y2, the carrier of X, REAL, L, addreal);
            then |.setopfunc(Y1, the carrier of X, REAL, L, addreal)-r.| <=
|.setopfunc(Y0, the carrier of X, REAL, L, addreal) - r.| + |.setopfunc(Y2,
            the carrier of X, REAL, L, addreal).| by ABSVALUE:9;
            then
A100:        |.x - setopfunc(Y1, the carrier of X, REAL, L, addreal) .|
<= |.setopfunc(Y0, the carrier of X, REAL, L, addreal) - r.| + |.setopfunc(
            Y2, the carrier of X, REAL, L, addreal).| by UNIFORM1:11;
            |.setopfunc(Y2, the carrier of X, REAL, L, addreal).| < 1/(
            i+1) by A40,A90,A97,A98;
            then |.setopfunc(Y2, the carrier of X, REAL, L, addreal).| < e/2
            by A87,XXREAL_0:2;
            then |.setopfunc(Y0, the carrier of X, REAL, L, addreal) - r.| +
|.setopfunc(Y2, the carrier of X, REAL, L, addreal).| < e/2 + e/2 by A92,
XREAL_1:8;
            hence |.r - setopfunc(Y1, the carrier of X, REAL, L, addreal).| <
            e by A100,XXREAL_0:2;
          end;
        end;
        hence |.r - setopfunc(Y1, the carrier of X, REAL, L, addreal).| < e;
      end;
      hence thesis by A89,A90;
    end;
    hence thesis;
  end;
  hence thesis by BHSP_6:def 6;
end;
