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

theorem Th26:
  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 & n <> 0 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 that
A1: d1 is_common_for_dom f and
A2: n<> 0;
  let m;
  assume
A3: m in dom (f|n);
    set G = (f|n).m;
    per cases;
    suppose
      n>=len f;
      then f|n = f by Lm1;
      hence thesis by A1,A3;
    end;
    suppose
A4:   n<len f;
      0+1<=n by A2,NAT_1:13;
      then
A5:   n in dom f by A4,FINSEQ_3:25;
      dom(f|n) = Seg len(f|n) & len(f|n) = n by A4,FINSEQ_1:59,def 3;
      then G = f.m & m in dom f by A3,A5,RFINSEQ:6;
      hence thesis by A1;
    end;
end;
