reserve
  a for natural Number,
  k,l,m,n,k1,b,c,i for Nat,
  x,y,z,y1,y2 for object,
  X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for FinSequence;
reserve D for set;

theorem
  i > 0 & i+m in Seg (n+m) implies i in Seg n & i in Seg (n+m)
  proof
  assume that
A1: i > 0 and
A2: i+m in Seg (n+m);
  1 = 0+1;
  then
A3: 1 <= i by A1,NAT_1:13;
A4: i+m <= n+m by A2,Th1;
  i <= n by A2,Th1,XREAL_1:6;
  hence i in Seg n by A3;
  i <= i+m by NAT_1:11;
  then i <= n+m by A4,XXREAL_0:2;
  hence thesis by A3;
end;
