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 Th39:
  X is included_in_Seg & k < l & 1 <= n & m <= len Sgm X & Sgm(X).m = k &
  Sgm(X).n = l implies m < n
proof
  assume that
A1: X is included_in_Seg and
A2: k < l and
A3: 1 <= n and
A4: m <= len(Sgm X) and
A5: Sgm(X).m = k and
A6: Sgm(X).n = l and
A7: not m < n;
  n < m by A2,A5,A6,A7,XXREAL_0:1;
  hence thesis by A1,A2,A3,A4,A5,A6,FINSEQ_1:def 14;
end;
