reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem
  x in X & y in dom f implies (X --> f)..(x,y) = f.y
proof
A1: x in X implies (X --> f).x = f by FUNCOP_1:7;
  dom (X --> f) = X;
  hence thesis by A1,FUNCT_5:38;
end;
