
theorem Th4:
B-Meas is sigma_finite
proof
    deffunc S(Nat) = In([.-$1,$1.],bool REAL);
    consider E be Function of NAT,bool REAL such that
A1: for i being Element of NAT holds E.i = S(i) from FUNCT_2:sch 4;

    reconsider E as SetSequence of REAL;
A2: for n being Nat holds E.n =[.-n,n.]
    proof
     let n be Nat;
     n is Element of NAT by ORDINAL1:def 12; then
     E.n =In([.-n,n.],bool REAL) by A1;
     hence thesis;
    end;

    for n being Nat holds E.n in Borel_Sets
    proof
     let n be Nat;
     E.n =[.-n,n.] by A2;
     hence E.n in Borel_Sets by MEASUR10:5;
    end; then
    reconsider E as Set_Sequence of Borel_Sets by MEASURE8:def 2;

A3: for n being Nat holds (B-Meas).(E.n) < +infty
    proof
     let n be Nat;
     -n in INT by INT_1:def 1; then
     -n in REAL & n in REAL by ORDINAL1:def 12,NUMBERS:15,19; then
     reconsider a= -n, b= n as R_eal by NUMBERS:31;
     E.n =[.a,b.] by A2; then
     (B-Meas).(E.n) = diameter([.a,b.]) by MEASUR12:71; then
     (B-Meas).(E.n) = b-a by  MEASURE5:6;
     hence (B-Meas).(E.n) < +infty by XREAL_0:def 1,XXREAL_0:9;
    end;

    now let z be object;
     assume z in REAL; then
     reconsider r=z as Real;
     consider n be Nat such that
A4:   |.r.| < n by SEQ_4:3;
     -n <= r & r <= n by ABSVALUE:5,A4; then
     r in [.-n,n.]; then
A5:  r in E.n by A2;
     dom E = NAT by FUNCT_2:def 1; then
     E.n in rng E by FUNCT_1:3,ORDINAL1:def 12;
     hence z in union rng E by A5,TARSKI:def 4;
    end; then
    REAL c= union rng E; then
    Union E = REAL by CARD_3:def 4;
    hence thesis by A3,MEASUR11:def 12;
end;
