reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

theorem Th5:
  i in dom D & i<>1 implies i-1 in dom D & D.(i-1) in A & i-1 in NAT
proof
  assume that
A1: i in dom D and
A2: i<>1;
  consider j be Nat such that
A3: dom D = Seg j by FINSEQ_1:def 2;
  i<>0 by A1,A3,FINSEQ_1:1; then
A4: i is non trivial by A2,NAT_2:def 1;
  then consider l being Nat such that
A5: i=2+l by NAT_1:10,NAT_2:29;
  reconsider l as Element of NAT by ORDINAL1:def 12;
  i >= 2 by A4,NAT_2:29;
  then
A6: 1+1-1 <= i-1 by XREAL_1:9;
  i<=j by A1,A3,FINSEQ_1:1;
  then
A7: i-1<=j-0 by XREAL_1:13;
A8: rng D c= A by Def1;
A9: l+1=i-(2-1) by A5;
  then i-1 in dom D by A3,A6,A7,FINSEQ_1:1;
  then D.(i-1) in rng D by FUNCT_1:def 3;
  hence thesis by A3,A6,A7,A9,A8,FINSEQ_1:1;
end;
