reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem
  p has_onlyone_value_in k & i<>k implies p.i=0
proof
  assume that
A1: p has_onlyone_value_in k and
A2: i<>k;
  per cases;
  suppose
    not i in dom p;
    hence thesis by FUNCT_1:def 2;
  end;
  suppose
    i in dom p;
    hence thesis by A1,A2;
  end;
end;
