reserve n,m for Nat,
  r,r1,r2,s,t for Real,
  x,y for set;

theorem Th31:
  for D be non empty set, d be Element of D, f be FinSequence of
  PFuncs(D,REAL), R be FinSequence of REAL st d is_common_for_dom f holds d
  is_common_for_dom R(#)f
proof
  let D be non empty set, d be Element of D, f be FinSequence of PFuncs(D,REAL
  ), R be FinSequence of REAL;
  assume
A1: d is_common_for_dom f;
  set m = min(len R,len f);
  let n;
  assume
A2: n in dom (R(#)f);
    set G = (R(#)f).n;
  len(R(#)f) = m by Def7;
  then m<=len f & n<=m by A2,FINSEQ_3:25,XXREAL_0:17;
  then
A3: n<=len f by XXREAL_0:2;
  1<=n by A2,FINSEQ_3:25;
  then
A4: n in dom f by A3,FINSEQ_3:25;
  then reconsider F=f.n as Element of PFuncs(D,REAL) by FINSEQ_2:11;
A5: d in dom F by A1,A4;
  reconsider r=R.n as Real;
  G=r(#)F by A2,Def7;
  hence thesis by A5,VALUED_1:def 5;
end;
