reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;

theorem Th43:
  f <> {} implies f/.1..f = 1
proof
  assume f <> {};
  then
A1: 1 in dom f by FINSEQ_5:6;
  f/.1 in {f/.1} by TARSKI:def 1;
  then
A2: 1 in (f qua Relation of NAT,D)"{f/.1} by A1,PARTFUN2:26;
  f"{f/.1} c= dom f by RELAT_1:132;
  then f"{f/.1} c= Seg len f by FINSEQ_1:def 3;
  then
A3: f"{f/.1} is included_in_Seg;
  thus f/.1..f = Sgm(f"{f/.1}).1 by FINSEQ_4:def 4
    .= 1 by A3,A2,Th1;
end;
