
theorem Th25:
for a be Real holds
 ex E be SetSequence of L-Field st
  (for n be Nat holds E.n = [.a,a+n.]) &
  E is non-descending & E is convergent & Union E = [. a,+infty .[
proof
    let a be Real;
    deffunc F(Element of NAT) = [.a,a+$1 .];
    consider E be Function of NAT, bool REAL such that
A1:  for n be Element of NAT holds E.n = F(n) from FUNCT_2:sch 4;

    now let x be object;
     assume x in rng E; then
     consider n be object such that
A2:   n in dom E & E.n = x by FUNCT_1:def 3;
     reconsider n as Element of NAT by A2;
     E.n = [.a,a+n.] by A1;
     hence x in L-Field by A2,MEASUR10:5,MEASUR12:75;
    end; then
    rng E c= L-Field; then
    reconsider E as SetSequence of L-Field by RELAT_1:def 19;
    take E;
    thus for n be Nat holds E.n = [.a,a+n.]
    proof
     let n be Nat;
     n is Element of NAT by ORDINAL1:def 12;
     hence thesis by A1;
    end;
    hence E is non-descending & E is convergent & Union E = [.a,+infty.[
     by Lm5;
end;
