reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;
reserve X for non empty set,
  Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function
  of Y,X,
  x,x1,x2 for Element of X;
reserve y for Element of Y;
reserve Y for non empty set,
  F for BinOp of X,
  f for Function of Y,X,
  x for Element of X,
  y for Element of Y;
reserve a,b,c for set;
reserve x,y,z for object;

theorem
  for f being constant Function holds f = (dom f) --> the_value_of f
proof
  let f be constant Function;
  thus dom ((dom f) --> the_value_of f) = dom f;
  let x be object;
  assume
A1: x in dom f;
  then f <> {} & ((dom f) --> the_value_of f).x = the_value_of f by Th7;
  hence thesis by A1,FUNCT_1:def 12;
end;
