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

theorem
  for D,C be non empty set, f be FinSequence of PFuncs(D,C), d be
  Element of D, n be Nat st d is_common_for_dom f holds d
  is_common_for_dom f /^ n
proof
  let D1,D2 be non empty set, f be FinSequence of PFuncs(D1,D2), d1 be Element
  of D1, n;
  assume
A1: d1 is_common_for_dom f;
  let m;
  set fn = f /^ n;
  assume
A2: m in dom fn;
  set G = fn.m;
  now
    per cases;
    case
      len f<n;
      hence thesis by A2,RELAT_1:38,RFINSEQ:def 1;
    end;
    case
A3:   n<=len f;
      1<=m & m<=m+n by A2,FINSEQ_3:25,NAT_1:11;
      then
A4:   1<=m+n by XXREAL_0:2;
A5:   m<=len fn by A2,FINSEQ_3:25;
      len fn = len f - n by A3,RFINSEQ:def 1;
      then m+n<=len f by A5,XREAL_1:19;
      then
A6:   m+n in dom f by A4,FINSEQ_3:25;
      G= f.(m+n) by A2,A3,RFINSEQ:def 1;
      hence thesis by A1,A6;
    end;
  end;
  hence thesis;
end;
