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 Th31:
  i in dom D1 & D1 <= D2 implies ex j st j in dom D2 & D1.i=D2.j
proof
  assume i in dom D1; then
A1: D1.i in rng D1 by FUNCT_1:def 3;
  assume D1 <= D2;
  then rng D1 c= rng D2;
  then consider x1 being object such that
A2: x1 in dom D2 and
A3: D1.i=D2.x1 by A1,FUNCT_1:def 3;
  reconsider x1 as Element of NAT by A2;
  take x1;
  thus thesis by A2,A3;
end;
