reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;

theorem
  for f be with_the_same_dom Functional_Sequence of X,REAL, F be
SetSequence of S, r be Real st (for n be Nat holds F.n = dom(
f.0) /\ great_eq_dom(f.n,r)) holds meet rng F = dom(f.0) /\ great_eq_dom(inf f,
  r)
proof
  let f be with_the_same_dom Functional_Sequence of X,REAL, F be SetSequence
  of S, r be Real;
  set E = dom(f.0);
  assume
A1: for n be Nat holds F.n = dom(f.0) /\ great_eq_dom(f.n,r);
  now
    let x be object;
    assume
A2: x in meet rng(F qua SetSequence of X);
    then reconsider z=x as Element of X;
A3: F.0 = E /\ great_eq_dom(f.0,r) by A1;
    F.0 in rng F by FUNCT_2:4;
    then x in F.0 by A2,SETFAM_1:def 1;
    then
A4: x in E by A3,XBOOLE_0:def 4;
    then
A5: x in dom inf f by MESFUNC8:def 3;
A6: now
      let n be Element of NAT;
      F.n in rng F by FUNCT_2:4;
      then
A7:   z in F.n by A2,SETFAM_1:def 1;
      F.n = E /\ great_eq_dom(f.n,r) by A1;
      then x in great_eq_dom(f.n,r) by A7,XBOOLE_0:def 4;
      then r <= (f.n).z by MESFUNC1:def 14;
      hence r <= (R_EAL(f#z)).n by SEQFUNC:def 10;
    end;
    for p be ExtReal holds p in rng R_EAL(f#z) implies r <= p
    proof
      let p be ExtReal;
      assume p in rng R_EAL(f#z);
      then ex n be object st n in NAT & (R_EAL(f#z)).n = p by FUNCT_2:11;
      hence thesis by A6;
    end;
    then r is LowerBound of rng R_EAL(f#z) by XXREAL_2:def 2;
    then r <= inf rng R_EAL(f#z) by XXREAL_2:def 4;
    then r <= (inf f).x by A5,Th2;
    then x in great_eq_dom(inf f,r) by A5,MESFUNC1:def 14;
    hence x in E /\ great_eq_dom(inf f,r) by A4,XBOOLE_0:def 4;
  end;
  then
A8: meet rng F c= E /\ great_eq_dom(inf f,r);
  now
    let x be object;
    assume
A9: x in E /\ great_eq_dom(inf f,r);
    then reconsider z=x as Element of X;
A10: x in E by A9,XBOOLE_0:def 4;
    x in great_eq_dom(inf f,r) by A9,XBOOLE_0:def 4;
    then
A11: r <= (inf f).z by MESFUNC1:def 14;
    now
      let y be set;
      assume y in rng F;
      then consider n be object such that
A12:  n in NAT and
A13:  y=F.n by FUNCT_2:11;
      reconsider n as Element of NAT by A12;
A14:  x in dom (f.n) by A10,MESFUNC8:def 2;
      x in dom inf f by A10,MESFUNC8:def 3;
      then
A15:  (inf f).z = inf rng R_EAL(f#z) by Th2;
      (f.n).z = (R_EAL(f#z)).n by SEQFUNC:def 10;
      then (f.n).z >= inf rng R_EAL(f#z) by FUNCT_2:4,XXREAL_2:3;
      then r <= (f.n).z by A11,A15,XXREAL_0:2;
      then
A16:  x in great_eq_dom(f.n,r) by A14,MESFUNC1:def 14;
      F.n = E /\ great_eq_dom(f.n,r) by A1;
      hence x in y by A10,A13,A16,XBOOLE_0:def 4;
    end;
    hence x in meet rng F by SETFAM_1:def 1;
  end;
  then E /\ great_eq_dom(inf f,r) c= meet rng F;
  hence thesis by A8;
end;
