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;

theorem
  for g being Function of X,X holds F.:(id X, g).x = F.(x,g.x)
proof
  let g be Function of X,X;
  thus F.:(id X, g).x = F.(id X.x, g.x) by Th37
    .= F.(x,g.x);
end;
