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
  0 < n & n < m implies Sgm{n,m} = <* n,m *>
proof
  assume that
A1: 0 < n and
A2: n < m;
A3: m in Seg m by A2,FINSEQ_1:3;
A4: n in Seg n by A1,FINSEQ_1:3;
  Seg n c= Seg m by A2,FINSEQ_1:5;
  then {n,m} c= Seg m by A3,A4,ZFMISC_1:32;
  then
a5: {n,m} is included_in_Seg by FINSEQ_1:def 13;
  then
A6: Sgm{n,m} is one-to-one;
A7: len(Sgm{n,m}) = card{n,m} by a5,Th37
    .= 2 by A2,CARD_2:57;
  then
A8: dom(Sgm{n,m}) = {1,2} by FINSEQ_1:2,def 3;
  then
A9: 1 in dom(Sgm{n,m}) by TARSKI:def 2;
A10: 2 in dom(Sgm{n,m}) by A8,TARSKI:def 2;
A11: rng(Sgm{n,m}) = {n,m} by a5,FINSEQ_1:def 14;
  then
A12: Sgm{n,m}.2 in {n,m} by A10,FUNCT_1:def 3;
A13: Sgm{n,m}.1 in {n,m} by A9,A11,FUNCT_1:def 3;
  now
    per cases by A13,A12,TARSKI:def 2;
    suppose
      Sgm{n,m}.1 = n & Sgm{n,m}.2 = n;
      hence Sgm{n,m}.1 = n & Sgm{n,m}.2 = m by A9,A10,A6;
    end;
    suppose
      Sgm{n,m}.1 = m & Sgm{n,m}.2 = m;
      hence Sgm{n,m}.1 = n & Sgm{n,m}.2 = m by A9,A10,A6;
    end;
    suppose
      Sgm{n,m}.1 = n & Sgm{n,m}.2 = m;
      hence Sgm{n,m}.1 = n & Sgm{n,m}.2 = m;
    end;
    suppose
      Sgm{n,m}.1 = m & Sgm{n,m}.2 = n;
      hence Sgm{n,m}.1 = n & Sgm{n,m}.2 = m by A2,a5,A7,FINSEQ_1:def 14;
    end;
  end;
  hence thesis by A7,FINSEQ_1:44;
end;
