reserve x,y for object,X for set,
  f for Function,
  R,S for Relation;
reserve e1,e2 for ExtReal;
reserve s,s1,s2,s3 for sequence of X;
reserve XX for non empty set,
        ss,ss1,ss2,ss3 for sequence of XX;
reserve X,Y for non empty set,
  Z for set;
reserve s,s1 for sequence of X,
  h,h1 for PartFunc of X,Y,
  h2 for PartFunc of Y ,Z,
  x for Element of X,
  N for increasing sequence of NAT;
reserve i,j for Nat;

theorem Th23:
  for X being set for s being constant sequence of X holds s.i = s.j
proof
  let X be set;
  let s be constant sequence of X;
  per cases;
  suppose
A1: X is empty;
    then s.i = {};
    hence thesis by A1;
  end;
  suppose X is not empty;
    then dom s = NAT by FUNCT_2:def 1;
    then i in dom s & j in dom s by ORDINAL1:def 12;
    hence thesis by FUNCT_1:def 10;
  end;
end;
