reserve X for non empty set,
  F for with_the_same_dom Functional_Sequence of X, ExtREAL,
  seq,seq1,seq2 for ExtREAL_sequence,
  x for Element of X,
  a,r for R_eal,
  n,m,k for Nat;

theorem Th6:
  for n be Element of NAT st x in dom inf(F^\n) holds (inf(F^\n)).x
  =inf((F#x)^\n)
proof
  let n be Element of NAT;
  now
    reconsider g=F as sequence of PFuncs(X,ExtREAL);
    let k be Element of NAT;
    ((F^\n)#x).k =((F^\n).k).x by MESFUNC5:def 13;
    then ((F^\n)#x).k = (g.(n+k)).x by NAT_1:def 3;
    then ((F^\n)#x).k =(F#x).(n+k) by MESFUNC5:def 13;
    hence ((F^\n)#x).k = ((F#x)^\n).k by NAT_1:def 3;
  end;
  then
A1: (F^\n)#x = (F#x)^\n by FUNCT_2:63;
  assume x in dom inf(F^\n);
  hence thesis by A1,MESFUNC8:def 3;
end;
