reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem Th20:
  for y being object st Seg k = {y} holds k = 1 & y = 1
proof let y be object;
  assume
A1: Seg k = {y};
  now
    per cases;
    suppose
      k = 0;
      hence thesis by A1;
    end;
    suppose
      k <> 0; then
A2:   k in Seg k by FINSEQ_1:3;
      then 1 <= k by FINSEQ_1:1;
      then Seg 1 c= Seg k by FINSEQ_1:5;
      then Seg 1 = {y} by A1,ZFMISC_1:33;
      hence thesis by A1,A2,FINSEQ_1:2,TARSKI:def 1,ZFMISC_1:3;
    end;
  end;
  hence thesis;
end;
