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

theorem Th1:
  for X being included_in_Seg set st 1 in X holds (Sgm X).1 = 1
proof
  let X be included_in_Seg set such that
A2: 1 in X;
  Sgm X <> {} by A2,FINSEQ_1:51;
  then len Sgm X >= 1 by NAT_1:14;
  then 1 in dom Sgm X by FINSEQ_3:25;
  then
A3: (Sgm X).1 in rng Sgm X by FUNCT_1:def 3;
  reconsider k1 = (Sgm X).1 as Nat;
  assume
A4: (Sgm X).1 <> 1;
A5: rng Sgm X = X by FINSEQ_1:def 14;
  then consider x being object such that
A6: x in dom Sgm X and
A7: (Sgm X).x = 1 by A2,FUNCT_1:def 3;
  ex k being Nat st X c= Seg k by FINSEQ_1:def 13;
  then
A8: k1 >= 1 by A5,A3,FINSEQ_1:1;
  reconsider j = x as Nat by A6;
  j >= 1 by A6,FINSEQ_3:25;
  then
A9: 1 < j by A7,A4,XXREAL_0:1;
  j <= len Sgm X by A6,FINSEQ_3:25;
  hence contradiction by A7,A8,A9,FINSEQ_1:def 14;
end;
