reserve n,k for Nat,
  X for non empty set,
  S for SigmaField of X;

theorem Th16:
  for f be with_the_same_dom Functional_Sequence of X,ExtREAL,
      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,ExtREAL,
      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 Def3;
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 <= (f#z).n by MESFUNC5:def 13;
    end;
    now
      let s be ExtReal;
      assume s in rng (f#z);
      then ex k be object st k in NAT & s=(f#z).k by FUNCT_2:11;
      hence r <= s by A6;
    end;
    then r is LowerBound of rng (f#z) by XXREAL_2:def 2;
    then r <= inf (f#z) by XXREAL_2:def 4;
    then r <= (inf f).x by A5,Def3;
    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,Def2;
      x in dom inf f by A10,Def3;
      then
A15:  (inf f).z = inf(f#z) by Def3;
A16:  x in E by A9,XBOOLE_0:def 4;
      (f.n).z=(f#z).n by MESFUNC5:def 13;
      then inf (f#z)<= (f.n).z by RINFSUP2:23;
      then r <= (f.n).z by A11,A15,XXREAL_0:2;
      then
A17:  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 A13,A16,A17,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,XBOOLE_0:def 10;
end;
